in this video i'm going to go over some basic problems associated in a typical course of geometry so what i recommend is that you pause the video and work on each of these multiple choice problems that's the best way you can learn so let's start with this one if the measures of angles abd and cbd are 5x plus 18 and 7x respectively which of the following could be the measure of angle abd now we're told that angle abc is a right angle so what that means is that angle abd and angle cbd they must be complementary they must add up to 90. so we can say that the measure of angle abd plus the measure of angle cbd adds up to 90 degrees now abd is equal to 5x plus 18 and cbd that's 7x so this has to add up to 90. so let's go ahead and combine like terms 5x plus 7x is 12x so we have 12x plus 18 is equal to 90. now let's subtract both sides by 18. 90 minus 18 that's going to be 72 now let's divide both sides by 12. 72 divided by 12 is 6. so now we have the value of x and our goal is to determine the measure of angle abd now we know that angle abd is equal to 5x plus 18. so now that we have the value of x all we need to do is plug it in to the equation so it's going to be 5 times 6 plus 18. 5 times 6 is 30 30 plus 18 is 48 so this is the measure of angle abd it's 48 degrees which means that c is the right answer number two c is the midpoint of segment a d if segment bc is equal to four and a d is 24 what is the length of segment a b according to the figure shown below so we know that a d is 24 units long and c is the midpoint of a d which means that c d has to be 12 and ac has to be 12. now bc is equal to 4. so what is a b well a b has to be the difference between ac and bc so 12 minus 4 is 8. a b and b c has to add up to ac so a plus 4 is 12. therefore this is the answer c is the right answer choice a b is equal to eight number three the measures of angle a b d and c b d are 10 x plus 20 and x squared minus 11 respectively according to the figure shown below what is the measure of angle cbd so go ahead and try that problem so angle abd that's given to us that's 10x plus 20. and angle cbd is equal to x squared minus 11. now abc is a straight line you could assume that if it looks straight so therefore these two angles have to add up to 180 which means that they're supplementary so we can say that 10x plus 20 plus x squared minus 11 must equal 180 so let's write this in standard form x squared plus 10x and we can combine 20 and negative 11. 20 minus 11 is 9. that's equal to 180 now let's subtract both sides by 180 so 9 minus 180 that's a negative 171. now what do you think we need to do to calculate the value of x at this point we need to find two numbers that multiply to negative 171 but that add to positive 10. we need to factor this expression two numbers that multiply to 171 are 9 and 19. now for these two numbers to add up to 10 we need to use negative 9 and positive 19. so to factor the expression it's going to be x minus nine times x plus 19 which means that if we set each factor equal to zero if we set x minus 9 equal to 0 and x plus 19 equal to 0 we can see that x can be equal to 9 or negative 19. now if x is equal to negative 19 this term 10x plus 20 or that expression that's going to have a negative value which we don't want so we're going to get rid of that answer so therefore x is equal to 9. now that we have the value of x we can determine the measure of angle cbd and we know it's x squared minus 11. so that's going to be 9 squared minus 11 and nine squared or nine times nine is eighty-one eighty-one minus eleven is seventy so the measure of angle cbd that's equal to seventy degrees which means b is the right answer number four the measures of the three angles of triangle abc have the ratio of five seven eight what is the difference between the measures of the largest and smallest angle in this triangle so let's draw a picture first so let's call this a b and c so the measures of the three angles have the ratio 5 578 so we can say that this is 5x 7x and 8x now the three angles of any triangle has to add up to 180. so we could say that 5x plus 7x plus eight x must equal to one eighty now five plus seven is twelve and twelve plus eight is twenty so twenty x is equal to one eighty and now we just need to divide both sides by 20. so 180 divided by 20 we can cancel the zero so it's 18 divided by two therefore x is equal to nine now the smallest angle is going to be angle a5x so 5 times 9 is 45 so that's the measure of angle a the largest angle is angle c which is 8x and 8 times 9 is 72 degrees now the difference between the largest angle and the smallest angle that's going to be 72 minus 45 and so that comes out to be 27 degrees and that's the answer which correlates to answer choice c number five point b is the center of the circle if the circumference of the circle is 20 pi units then what is the area of the shaded region now since b is the center of the circle that means that a b and b c represents the radius of the circle the distance between the center of the circle and any point on a circle is the radius of the circle so those two are the same now we need to calculate r and we have the circumference which is two pi r the circumference in this problem is 20 pi and if we divide both sides by 2 pi we can get the value of r on the left side we can cancel pi and 20 divided by 2 is 10 so the radius is 10 units long now to calculate the area of the shaded region you need to realize that it's the difference between the area of the larger object which is the area of the circle minus the area of the small object which is the triangle now the area of a circle is pi r squared and the area of a right triangle is one half base times height we already know the radius it's 10 and the base of the triangle that's bc that's 10 and the height of the triangle which is a b that's also 10. ten squared is a hundred half of ten is five and five times ten is fifty so this is the answer it's one hundred pi minus fifty square units now if you want to get the decimal value for that it's equal to 264.16 square units so that's the area of the shader region number six what is the area of an equilateral triangle with a side length of twelve so let's draw a picture first an equilateral triangle is a triangle where all three sides are the same so in this case each side is equal to 12. it's also an equal angular triangle where all the angles are the same so they all have to be 60 degrees because all three angles of a triangle must add up to 180 but we don't need the angle for this particular problem now there is a specific formula that you must know to calculate the area of an equilateral triangle and here it is it's the square root of 3 over 4 times s squared where s is the side length of the triangle so for this problem is the square root of three over four times twelve squared twelve times twelve is one hundred and forty four and one forty four divided by four is thirty six so the answer is 36 square root 3 which means a is the right answer choice number seven what is the value of x in degrees in the figure shown below now before we start this problem i want to go over some basics that you need to know now we're told that l is parallel to m so these two lines are parallel to each other so we can write this symbol to represent that now anytime you see this double line that means that the two lines are parallel if you see this symbol it means that the lines are perpendicular which means that they intersect at right angles that's a 90 degree angle symbol so let's say if we have two parallel lines cut by a transversal and let's call this one two three four five six seven eight now you need to know that the alternate exterior angles are congruent so one and eight are alternate exterior angles they're on the outside or the exterior of the parallel lines and they're on alternate sides of the transversal so one and eight have the same measure two and seven are also alternate exterior angles so two equals seven now the next thing you need to be familiar with are alternate interior angles so three and six are alternate interior angles they're on the interior of the parallel lines and they still exist on alternate sides of the transversal four and five are also alternate interior angles so angle four equals angle five now four and six are known as same side consecutive interior angles and their supplementary so angle 4 plus angle 6 adds up to 180 degrees now angles 3 and 5 are also same side consecutive interior angles now 7 and 6 are known as vertical angles and they're the same so 7 equals 6 and the measure of angle 5 is equal to angle 8. now 2 and 3 are vertical angles and 1 and 4 are also vertical angles the next type of angle that you need to know is the corresponding angle so 2 and 6 are corresponding angles they have the same measure they're equal to each other 1 and 5 are corresponding and 3 corresponds to seven and four corresponds to eight so those are some things that you want to keep in mind when dealing with parallel lines and transversals now let's focus on this particular problem so these two angles are vertical angles which means that this one has to be 70 and this has to be 60 because those two are congruent now what do you think this angle is at this point notice that these two angles form a straight line which means that they're supplementary they add up to 180. 180 minus 70 is 110 so this is 110 degrees which means this is also 110 and 180 minus 60 is 120. now these two are corresponding to each other so this has to be 120 which means this is 120 and this is 60 and these two are vertical angles so this one has to be 60 as well now these two angles are alternate exterior angles so they're the same which means this has to be 70 so that's 70 and this is 110 and so forth so now we can calculate x let's focus on the triangle so this is x this is 70 and this is 60. now the three angles of a triangle we know has to add up to 180 so this will help us to calculate the value of x 70 and 60 adds up to 130 and to calculate x we need to subtract both sides by 130 so x is going to be 180 minus 130 which is 50 degrees and that is the answer so now you had a basic review on alternate interior angles exterior angles transversals parallel lines and things like that let's move on to the next question by the way for those of you who want more help in geometry check out my geometry video playlist on youtube and you could find a specific topics that you might need help in so let's finish this problem how many diagonals does a regular hexagon have first let's draw a picture so here's a regular hexagon but my picture doesn't look that great so let's do this one more time now there's a formula that you can use to calculate this answer and it's n times n minus three divided by two so this equation tells you the number of diagonals a particular polygon contain so we have a polygon with six sides so n is six so it's going to be six times six minus three divided by two six minus three is three and six times three is eighteen eighteen divided by two is nine so the answer is nine now let's go ahead and show this so let's start with this side a diagonal is basically a line that connects one side to the other side basically it connects from one vertex to another vertex but it can't be like a consecutive vertex so you can't just draw a line here that would be a side instead of a diagonal so this is one diagonal this is another one so that's two diagonals and this one is another one that's three now let's start with this point so this is a diagonal that's four five six and then if we start with this point seven and eight and then the last one is from here to here this would be nine and that's it that's all we control we can't draw this one because we already have the blue line we can't draw that one we already have a line here so there's a total of nine diagonals that we can draw for this particular hexagon so b is the right answer number nine what is the value of x in the figure shown below now one way you could find the answer is by using the exterior angle theorem which states that the exterior angle is equal to the sum of the remote interior angles now the interior angles are the angles on the inside of the triangle so these are x interior angles on the inside the exterior angle is the angle on the outside so this is an exterior angle now the remote interior angles are the two interior angles that are away from this angle basically these two angles are the remote interior angles so they're not connected or associated with this angle they're far away from it but they're still on the inside of the triangle so they're still interior angles and so the exterior angle is the sum of the remote interior angles so x is equal to 40 plus 65 which is 105. now another way in which you can get the same answer is by calculating the value of angle acb because you know that the three angles of a triangle has to be oh has to add up to 180. so let's call this angle y so 40 plus 65 plus y has to equal 180 so 40 plus 65 is 105 and 180 minus 105 is 75 so that's the measure of angle y now x plus y has to add up to 180 because as you can see they form a straight angle they're supplementary now we know that y is 75 so x has to be 180 minus 75 which means x is still 105. so you have two ways in which you can get the same answer but if you know the exterior angle theorem you could simply see that x is the sum of the remote interior angles so therefore e is the right answer number 10 what is the measure of each interior angle inside a regular hexagon so once again let's draw a picture now a regular hexagon is a hexagon where all sides are the same so all sides are equal which means that every interior angle is equal so we need to calculate the value of x can we do so well there is an equation that will tell us the total the sum of the all of the interior angles inside this hexagon and that equation is this so s which will correlate for the sum is equal to 180 times n minus two for a hexagon n is six and six minus two is four and 180 times 4 that's 720 degrees so that is the sum of all six interior angles inside this particular hexagon so to find the value of each interior angle we need to take the sum and divide it by the number of sides so there are 6 sides 720 divided by 6 is 120 so that is the measure of each interior angle inside a regular hexagon it's 120 degrees 11 what is the measure of an exterior angle of a regular pentagon let's start with the picture so here is a pentagon and because you're dealing with a regular pentagon all sides are the same now granted my picture is not perfect so you just have to make do with it the exterior angle is the angle on the outside so in this case x is the exterior angle y is an interior angle and for regular pentagon all the interior angles are the same notice that the exterior angle and the interior angle they add up to 180 because they form a straight angle they're supplementary our goal is to calculate x now the most efficient way to calculate the exterior angle is to take 360 and divide it by n now a pentagon has five sides one two three four five so we need to divide 360 by five and so this will give you 72. so that is the final answer is the most simplest way to get to it which means that b is the right answer but now what if you didn't know about this formula what should you do well first you can calculate the measure of each exterior angle like we did in the last problem it's going to be 180 times n minus 2 this will give the sum of all of the interior angles and then divided by the number of sides and that will give you the measure of each interior angle inside a regular polygon so now let's go ahead and plug in n so n is 5 in this example now 5 minus 2 is 3 and 180 times 3 is 540. so then it's going to be 540 divided by 5. now 500 divided by 5 is 100 40 divided by 5 is 8 so this is going to be 108. so if y is 108 and x and y are supplementary x has to be 180 minus 108 which will give us 72. so that's another way in which you can get the same answer but as you can see this formula is a lot easier so b is the right answer in this problem number 12 what is the supplement of the angle shown below so go ahead and try this free response problem now we're given the angle in dms degrees minutes and seconds to calculate the supplement we know that two supplementary angles add up to 180 so the supplement is going to be 180 minus this particular angle so how can we subtract 180 by a value in dms we need to convert this angle to an angle in degrees minutes and seconds so how can we do that well first we need to borrow one degree from 180. so if we borrow one degree it's going to be 179 degrees and 60 minutes is equal to 1 degree so it's 179 degrees and 60 minutes now we need something with a value in the seconds so let's borrow a minute one minute is equal to 60 seconds so it's going to be 59 minutes because we took one away and 60 seconds so 180 degrees is equal to this value in dms now we can subtract these two values so first let's subtract the values in degrees 179 minus 112 is 67. 59 minus 32 is 27 and 60 minus 45 is 15. so this is the answer in dms it's 67 degrees 27 minutes and 15 seconds 13. which of the following triangles contain an altitude is it a b c or d the answer is a a contains an altitude an altitude connects from one vertex to the opposite side of a triangle and it meets that side at a right angle so a d is an altitude for triangle a that is for triangle in answer choice a so this is the answer b contains a perpendicular bisector now like in altitude a perpendicular bisector touches one side of the triangle at 90 degrees however it doesn't have to pass through vertex a at the same time the perpendicular bisector bisects the opposite side bc into two congruent parts which means that d is the midpoint of b c and b d and d c are equal to each other so that's the perpendicular bisector now for answer choice c b d is a median a median bisects the opposite side into two congruent parts so a g and d c are congruent and d is the midpoint of ac however unlike a perpendicular bisector it doesn't meet the opposite side of right angles it simply bisects the opposite sides into two equal parts now for answer choice d it contains an angle bisector dc is the angle bisector it bisects angle c into two equal parts so acd is equal to bcd so if acd is 30 bcd is 30 they will have the same measure so that's an angle bisector and so that's it for this problem so a is the correct answer a d is the altitude number 14 the endpoints of a diameter of a circle are 1 comma 2 and 7 comma 10. which of the following could be the equation of the circle to write the equation of the circle in standard form you need to use this formula it's x minus h squared plus y minus k squared is equal to r squared now the center is h comma k and r represents the radius of the circle now we have a circle and one point is one comma two and the other point is seven comma ten now these are the end points of the diameter of the circle and the diameter of the circle passes through the center in fact the center is the midpoint of these two points which are the endpoints of the diameter so we need to use the midpoint formula to get the midpoint that's going to give us the center and then we can use the distance formula to get the radius of the circle that is we have to use the distance formula between the midpoint and any endpoint of the diameter it can be 710 or it can be one comma two but we have to use the midpoint in order to calculate the radius so here is the midpoint formula it's the average of the x values and the average of the y values so what is the average of one and seven the average of one and seven is four four is the midpoint of one and seven the midpoint of two and ten is six but if you wanna use the formula here's what you need to do so we have the point one comma two and seven comma ten so this is going to be x one y one and this is x two and y two so x one is one x two is seven y one is two y two is ten so one plus seven is eight and two plus ten is twelve now eight divided by two is four twelve divided by two is six so this is the midpoint which is also the center so therefore the center of the circle is 4 comma 6. so we have the values of h and k now h is 4 k is 6. now let's calculate the radius so we need to use the distance formula between the center of the circle and also any endpoint i'm going to use this one 1 comma 2. so here's the distance formula it's the square root of x2 minus x1 squared plus y2 minus y1 squared so i'm gonna call this x1 and y1 that's one comma two and four and six this is going to be x2 and y2 now x two is four x one is one y two is six y one is two four minus one is three and six minus two is four three squared is nine four squared is sixteen and nine plus sixteen is twenty-five and the square root of twenty-five is five so the distance between the center and one of the endpoints of the diameter is five which means the radius of the circle is five so now let's use this formula so h is four k is six and r is five so five squared five times five that's 25. so this is the equation of the circle in its standard form which means a is the right answer number 15 what is the area of the scalene triangle shown below so we have a triangle with three sides and the three sides are not the same so it's not an equilateral triangle the three sides are different how can we calculate the area of this particular triangle we'll need to use huron's formula so first we need to calculate s which is one half of the perimeter now lower case a is across angle a and side b is across angle b and this is going to be side c so s is going to be the perimeter a plus b plus c divided by two so it's half of the perimeter a is seven b is eight c is nine now seven plus eight is fifteen fifteen plus 9 is 24 and 24 divided by 2 is 12. so s is equal to 12. that's the first step now the second step is to plug in into this formula so the area is going to be s times s minus a times s minus b times s minus c so that's heron's formula now s is 12. s minus a that's going to be 12 minus 7 s minus b that's 12 minus 8 and s minus c is 12 minus 9. now 12 minus 7 is 5. twelve minus eight is four and twelve minus three i mean twelve minus nine is three so this is what we have now let's see if we can simplify the square root so twelve we can break that into four times three now notice that we have two fourths which is sixteen the square root of sixteen is four so two fours will come out of the radical as one four and notice that we have two threes so that will come out the radical as a single three and so we have a five left over four times three is twelve so it's going to be twelve square root five so this is the area of the triangle which means answer choice a is the right answer number 16 the ratio of the length to the width of a rectangle is 8 to 5. if the area of the rectangle is 360 what is the perimeter of the rectangle so let's begin by drawing a picture and so let's say this is the length and this is the width now we know that the area is left times width that's the area of a rectangle and the area is 360. now if we could find the length in the width we can calculate the perimeter the perimeter is simply 2l plus 2w in order to solve for two variables we need two equations the second equation we can get based on the ratio so the ratio between the length and the width is eight to five so therefore l divided by w is eight over five now let's cross multiply or rather let's solve for l let's multiply both sides by w so l is 8w divided by 5. so now what i'm going to do is i'm going to replace l with that value so 8w over 5 times w so now i have one equation in terms of one variable so i can solve for w now let's multiply 5 to both sides so we can get rid of this fraction 5 times 360 that's 1800 so 1800 is equal to 8 times w squared now let's divide both sides by 8. 1800 divided by eight is two twenty-five and now let's take the square root of both sides the square root of 225 is fifteen so w is fifteen now let's use this equation to calculate the value of l so the area is 360 and w is 15. so l the length of the rectangle is 360 divided by 15 which is 24. so now that we have the length and the width we can calculate the perimeter so the length is 24 and the width of the rectangle is 15. the perimeter is the sum of all four sides it's 2l plus 2w so as we said before l is 24 and w is 15. 2 times 24 is 48 2 times 15 is 30 and 48 plus 30 is 78 so this is the perimeter of the rectangle it's 78 units long and that's it for this problem number 17 the length of the diagonal of the square is 10 inches what is the area of the square well let's see if we could come up with a relationship that relates the area of the square to its diagonal so here is the diagonal of the square let's call it d and let's call the side length x now the area of a square is simply length times width or x times x now what's the relationship between x and d well we can use the pythagorean theorem since we have a right triangle so let's say this is a b and c so c squared is equal to a squared plus a b squared in this example d is the same as c and a and b are both equal to x so 1x squared plus 1 x squared is 2x squared and then if we take the square root of both sides we can see that d is going to be the square root of 2 times x the square root of x squared is just x so that's the relationship between the length of the diagonal and the side length now the area is equal to x squared and if we go back a few steps notice that we have d squared in terms of x squared d squared is equal to x squared let's divide both sides by two so two over two is one so notice that one half d squared is equal to x squared and if x squared is equal to a that means a is equal to half d squared so this is the formula that we need this is how we can calculate the area of a square if we're given the length of the diagonal now this formula is actually very similar to the area for other shapes that we're going to go over later in this video so make sure you know this equation so the area is one half d squared so the length of the diagonal is 10 and 10 squared is a hundred and half of a hundred is fifty so the area is fifty square units and that's the answer which correlates to answer choice b 18 in rhombus a b c d b c is 10 and b e is 6. what is the area of the rhombus now in the previous problem we said that the area of a square is one half d squared it turns out that the area of a rhombus is very similar however the diagonal lengths of the rhombus are not the same bd doesn't have to equal ac so the area is going to be one half d1 times d2 as opposed to one half d times d which is equivalent to that formula so let's call this a d1 that's bd and ac is d2 so we need to calculate d1 and d2 before we can calculate the area of the rhombus so in this problem we're told that bc is 10 and be is six so how can we figure out everything else that we need it's important to know that the diagonals of a rhombus bisect each other at 90 degrees so this is a 90 degree angle the second thing is that all four sides of a rhombus are equal so all of these are equal to 10. the next thing you need to know is that a rhombus bisects the diagonals so what that means is that be is congruent to ed and ae is congruent to ec so e is the midpoint of b d and e is the midpoint of ae i mean ac so if b e is six e d is also six now we need to calculate ec and ae as well we have b e and e ed let's focus on the right triangle bec we can see that bc is 10 be a six what's easy so we can use the pythagorean theorem c squared is equal to a squared plus b squared c is the hypotenuse which we'll call 10 and let's say that we're looking for a a is ec b is going to be b e so b is six ten squared is a hundred six squared over six times six is thirty six and a hundred minus thirty six is sixty four and then let's take the square root of both sides the square root of 64 is 8. now there are some special right triangle numbers that you want to commit to memory so let me just get rid of some stuff the first is the 3 4 5 right triangle what this means is that 3 squared plus 4 squared is equal to 5 squared it follows the pythagorean theorem 9 plus 16 is 25. so that's one special right triangle that you want to know from this point forward the next one is the 5 12 13 triangle and notice that this side is eight if you multiply the three four five by two it gives you the six eight ten triangle so any ratio or any multiple of that ratio will also work so we have the 6 8 10 triangle in this example now some other numbers that you want to be familiar with is the 7 24 25 triangle and is also the 8 15 17 triangle and there's the 9 40 41 and also the 11 60 61 but these four are the most common ones that you encounter as for the rest you could simply use the pythagorean theorem to calculate the value of the missing side but now let's finish this problem so we have ac ec is 8 which means that ae is also eight now the left of the first diagonal d1 is going to be six plus six which is twelve and the length of the second diagonal d2 is going to be 8 plus 8 which is 16. and so now we can calculate the area of the rhombus it's one half of d1 times d2 so d1 in this example is 12 d2 is 16 and half of twelve is six six times sixteen well six times ten is sixty six times six is thirty six so sixty plus thirty six that's ninety six so the area of this particular rhombus is 96 square units which means c is the right answer number 19 in kite a b c d a d is 20 b e is 12 and b c is 13. what is the area of the kite so like a rhombus the area is the same so this is going to be d1 that's bd and ac is d2 the area is one-half d1 times d2 so let's focus on calculating the length of the two diagonals so a d is 20 b e is 12. bc is 13. now like a rhombus the diagonals of a kite they bisect each other at right angles now the second thing is that a b and a d are congruent which means that a b is 20 as well bc is congruent to dc so dc is also 13 and be is congruent to ed which means ed is 12. now we need to find the value of ec and ae we already have d1 we can see that it's 12 plus 12 which is 24. so we need to calculate d2 notice that we have a right triangle so we can use the pythagorean theorem to find the missing side so we have 12 and 13. what do you think is the missing side so from the last problem recall that 5 12 13 are three numbers of a right triangle so the missing side has to be five now some other right triangles that we mentioned were the 7 24 25 triangle the 3 4 5 triangle and the 8 15 17 triangle which of these triangles do you think will be useful to find a miss inside ae because this is also a right triangle is it the 7 24 25 the 3 4 5 the 8 15 17 or the 5 12 13. it turns out it's this one if we multiply the three four five triangle by four notice what happens so three times four is twelve four times four is sixteen five times four is twenty notice that we have two of the three sides we have twelve and twenty which means the missing side has to be sixteen and so that's why you wanna know these special numbers because it can save you a lot of time when solving these types of problems so we have the 12 16 20 triangle so now we can calculate d2 which is 16 plus 5 and that's 21. so the area is going to be one half d1 times d2 so d1 is 24 d2 is 21. now half of 24 is 12 so we're going to multiply 12 by 21 12 times 21 is 252 and so that's the area of this particular kite is 252 square units which means b is the answer number 20 what is the volume surface area and diagonal length of the rectangular prism shown below so we have the length which is 12 the width of the rectangular prism is 5 and the height is 4. to calculate the volume is simply length times width times height so it's going to be 12 times five times four now twelve times five is sixty sixty times four is two hundred forty so it's going to be two hundred forty cubic units or units cube now what about the surface area so we got to find the area of all six faces so first let's find the area of the bottom face which is length times width and also the top face is also length times width so the area for the bottom and the top we could say it's two times lw next we have this side the left side and also the right side which is basically the width times the height so it's going to be 2wh and then we have the area of the face on the back and the one on the front which has the dimensions l times h so it's going to be 2 lh so this is the formula to calculate the surface area of rectangular prism so let's go ahead and plug in the numbers so l is 12 w is 5 and then wh that's going to be 5 times 4 and then 2 lh 12 times 4. now we said that twelve times five is sixty and five times four is twenty and two times twelve is twenty four now two times sixty is one twenty two times twenty is forty and four times twenty four four times twenty is eighty four times four is sixteen so that's ninety six one twenty plus forty is one sixty and now let's add 160 and 96 6 and 9 is 15 1 and 1 is 2. so the surface area is 256 square units now how can we calculate the length of the diagonal so let's say if this is a b c d and this is e f g we want to find the distance between point c and point f how can we do that well let's design a formula that's going to help us to get the answer first let's draw a line between point c and point g so notice that let's call this z notice that we have a right triangle so we can say that z squared is equal to l squared plus w squared now notice that we have another right triangle between z f g and c f so let's call c f d the length of the diagonal so d squared is going to equal h squared plus z squared now i know what z squared is equal to z squared is l squared plus w squared so i'm going to replace c squared with that so therefore d squared is equal to h squared plus l squared plus w squared so the left of the diagonal is the square root of l squared plus w squared plus h squared and that's the formula that we need so let's plug in l is 12 w is 5 h is 4. 12 squared is 144 5 squared is 25 and 4 squared is 16. now 144 plus 16 that's 160 and 160 plus 25 is 185 so we have the square root of 185 which we cannot simplify but as a decimal that's about 13.6 if you round it to nearest tenth so that's the answer that's the diagonal length of the rectangular prism 21 triangle abc is similar to triangle def using the figures shown below what is the sum of x and y now if the triangles are similar that means that their sides are proportional so we can set up a proportion now this can help you in the first fraction i'm going to put the information for triangle one and for the second fraction triangle two now let's say the top part is going to be referenced to the long side and the bottom part the short side so the long side of triangle one that's bc which is 12. the short side of triangle one is a b which is eight the long side of triangle 2 that's x e f has to correspond to b c and the short side of triangle 2 is d e which is 20. so now let's solve for x let's cross multiply so this is going to be 8x that's equal to 12 times 20 which is 240. now let's divide both sides by eight two forty divided by eight is thirty so x is equal to that now let's talk about how you can confirm that 30 is the correct answer let's divide 20 by eight 20 divided by 8 is 2.5 so 2.5 is the scale factor if you take 8 and multiply by 2.5 you get 20. so therefore if we take 12 and multiply it by 2.5 we should get the same answer and 12 times 2.5 is 30. so that's a quick way to confirm if you have the right answer now let's calculate the value of y whenever two triangles are similar to each other the angles are congruent so angle a and angle d are congruent angle b and angle e they're both equal they're both 90 and then the other two angles angle c and angle f these two are congruent as well so therefore we can set 40 equal to 4y plus 12. so let's subtract both sides by 12. 40 minus 12 is 28 so 28 is equal to 4y and now let's divide by 4. so y is 7. so now we can calculate the sum of x and y it's 30 plus 7 or 37. so therefore d is the right answer number 22 b is the midpoint of ac and d is the midpoint of ce if bd is 4x minus 8 and ae is 5x minus 1 what is the length of segment bd so we're told that b is the midpoint of ac that means that a b and b c are the same and if d is the midpoint of c e c d and d e are congruent when we have the situation it turns out that bd is one half of segment ae now we know that bd is 4x minus 8 and ae is five x minus one so if we can calculate the value of x we can determine the length of segment bd so let's multiply both sides by two to get rid of the fraction one half times two is one now let's distribute the two two times four x is eight x two times negative eight is negative sixteen and so that's going to be equal to five x minus one now let's subtract both sides by five x and let's add 16 to both sides 8x minus 5x is 3x negative 1 plus 16 is 15. now let's divide both sides by three fifteen divided by three is five so x is equal to five and bd we know it's four x minus eight so let's replace x with five four times five is twenty and twenty minus eight is twelve so this is the length of segment bd is 12 units long so c is the right answer for this problem 23 what is the value of x in the trapezoid shown below now we can see that e is the midpoint of a b and f is the midpoint of c d since b e and a e are congruent and c f equals d f now when that happens this number 24 ef is the average of bc and ad so therefore 24 is going to be one half the value of x and 18. a quick way to determine the value of x is to look at the difference between 18 and 24. the difference is six so to go from b c to e f you have to add six and to go from e f to x or a d you have to add another six so x is going to equal 30. but let's use this equation to get the answer so first i'm going to multiply both sides by two to get rid of the one-half two times 24 is 48 next i'm going to subtract both sides by 18. 48 minus 18 is 30. so x is equal to 30 which means that c is the right answer 24 what is the measure of angle d in quadrilateral abcd so first we need to determine the sum of all of the interior angles of a quadrilateral a quadrilateral is basically a four-sided figure and to determine the sum of the interior angles we can use this formula so a quadrilateral has four sides so n is four four minus two is two so the sum of the interior angles has to be 360. so therefore we could say that angle a which is 5x plus 10 plus angle b which is 110 plus the measure of angle c that's x squared minus 44 plus the measure of angle d which is 7 x minus 4. all of that has to add to 360. so now let's calculate the value of x so let's write this in standard form we have x squared and let's combine the like terms 5x and 7x adds up to 12x and then we have 10 plus 110 that's 120 minus 44. 76 minus 4 that's 72 and that's equal to 360. now let's subtract 72 by 360. and so that's going to be negative 288 so that's what we now have now in order to solve for x we need to factor so what two numbers multiplies to negative 288 but add to the middle coefficient positive 12. well let's make a list if we divide this by four this will give us 72. one of these has to be negative and these two numbers they don't add up to 12. so let's try a bigger number let's try 8. 288 divided by 8 is 36 so that's not going to work let's try 12. negative 288 divided by negative 12 is 24. now this works these two numbers add up to 12. so to factor it's going to be x minus 12 times x plus 24 and that's equal to zero now let's set each factor equal to zero so we get two possible answers for x x could be positive 12 or negative 24. now we can't use this answer because angle a and d will be negative which is not possible so therefore x has to be 12. so now angle d which is equal to seven x minus four that's going to be seven times twelve minus four now seven times ten is seventy seven times two is fourteen so 7 times 12 is 84 and 84 minus 4 is 80. so this is the measure of angle d it's 80 degrees therefore d is the right answer choice 25 what is the value of x in the figure shown below so we can see that d e is equal to x a e is 16 and a b is 34. now if a b is 34 and ae is 16 eb has to be the difference between 16 and 34. so 34 minus 16 that's 18. now we have dc which is 36 and we have de but we don't have ec so we got to calculate ec so d e plus e c has to equal the total segment dc d e is x we're looking for ec and dc is 36 so therefore we can say that ec is 36 minus x if we subtract both sides by x now in order to calculate the value of x we need to use something called the chord chord power theorem and the formula for that is a b is equal to cd so let me put this in a different color this would be a this would be b this is c and this is d so a times b that's going to be 16 times 18 c times d that's 36 minus x times x now 16 times 18 that's 288 and then 36 times x is 36x and then negative x times x is negative x squared now i'm going to take everything from the right side move it to the left so on the left side this is going to be positive x squared and this will be negative 36 x and this will remain positive 288 so once again i need to factor this expression so what two numbers multiply to 288 but add to negative 36. this is going to be 12 and 24 but both have to be negative negative 12 times negative 24 is positive 288 but they add up to negative 36. so it's going to be x minus 12 and x minus 24 which means x can be equal to 12 or 24. it really doesn't matter 16 times 18 that's going to equal x which is 12 and then 36 minus x or 36 minus 12 is 24 or if you do it this way if x is 24 then 36 minus 24 is 12. so either way it will still work now since 24 is not listed as one of the answers we're going to go with 12. so therefore d is the right answer in this problem 26 given the points a b and c on the triangle shown below what is the slope of altitude bd now the first thing we need to know is that an altitude is perpendicular to the side that it interacts with that is opposite to the vertex from which it comes from now you need to be familiar with the slope of perpendicular lines so consider these two lines line l and line k and let's say that these lines are perpendicular to each other and let's say if you know the slope of line l let's say it's two over three what is the slope of line k which is perpendicular to line l to find the slope you need to flip the fraction and change the sign from positive to negative or vice versa so it's the negative reciprocal of the other line now to find the slope of bd we need to find the slope of the line that's perpendicular to it that is ac and we could find a slope of this line because we have two points on that line and so to calculate the slope between two points which would be the slope of ac we can use this formula y2 minus y1 divided by x2 minus x1 so this is going to be x1 that's y1 and this is x2 and y2 so y2 is nine y one is three x two is four x one is two nine minus three is six four minus two is two six over two is three so the slope of segment ac is three or three over one so the slope of bd which is perpendicular to ac it has to be one over three with a negative sign so it's negative one third which means that d is the right answer number 27 what is the difference between the values of x and y in the figure below now we're given that arc ae the measure is 110 and the measure of arc bd is 50. so with this information how can we calculate the difference between x and y well let's break up this problem into two parts the first part is a chord chord angle problem so this is a e b d and f so arc ae is 110 arc bd is 50. and we need to calculate x it turns out that x is the average of 50 and 110. so it's one half 50 plus 110. 110 plus 50 is 160 and half of 160 is 80. so as you can see 80 is the angle between 50 and 110 so that's the value of x now let's calculate the value of y so we have two secants passing through the circle a secant is a line that touches the circle at two points so we have secant ac and secant ec now ae is still 110 bd is still 50. now to calculate angle y it turns out that it's going to be one half the difference between the two arcs so the difference between them is going to be 110 minus 50 which is 60. so y is going to be half of 60 which is 30. so that's the measure of angle y which is the same as angle c now the difference between x and y 80 minus 30 is 50. so c is the right answer 28 what is the length of dc in the figure below so what do we need to do to this problem we need to use the secant tangent power theorem the formula for that is t squared is equal to e times s t represents the tangent segment and keep in mind a tangent line touches the circle only at one point a secant line touches it at two points e is the exterior portion of the secant segment so e is bc and s is the secant itself which is uh ac so t is equal to x and the exterior part is nine the secant is the sum of a b and b c so it's seven plus nine which is sixteen now let's multiply nine times sixteen nine times sixteen is 144 and so that's equal to x squared so all we have to do at this point is take the square root of both sides so the square root of 144 is 12. so therefore c is the right answer 29 the height and volume of a cone are 15 inches and 320 pi cubic inches respectively what is the total surface area of the cone so let's begin with a picture so here's the cone this is the radius of the cone here we have the height of the cone and this is the slant height and this is the right angle now the surface area the total surface area is going to be the area of the base plus the lateral area so the area of the base is basically the area of the circle on top which is pi r squared that's the area of the circle now the lateral area la that's going to be the area around the cone basically that wraps it around and it turns out that it's simply pi times r times l the lateral area if you want to like determine it it's one half the perimeter times the slant height the perimeter of a circle is basically the circumference of a circle which is two pi r so half of two pi it's just pi so it's pi times r times l so that's the formula that you need to calculate the surface area of a cone now let's go ahead and finish this problem so let's write down what we have so we know that the height is 15 inches the volume is 320 pi cubic inches what we need is the radius and the slide height so the volume of a cone is one third times the volume of the cylinder which is pi r squared times the height we don't have r h is 15 and volume is 320 pi so we could cancel pi one third of 15 is five so 320 is equal to 5r squared now let's divide both sides by 5. 320 divided by 5 is 64. so now we need to take the square root of both sides the square root of 64 is eight so the radius is eight cubic inches now let's calculate the slot height so let's focus on the right triangle that we have h is 15 r is eight so what's the missing side notice that this is the 8 15 17 triangle so now we have the slight height which is 17 inches so the surface area is going to be pi r squared where r is 8 plus pi r l where l is 17. 8 squared is 64. 8 times 17 is 136 and 160 i mean 64 plus 136 is 200 so the surface area of this particular cone is 200 pi square inches number 30 c d is perpendicular to a b if a b is 7 x minus 5 and a d is 3x minus 1 what is the length of segment cd if the radius of circle c is ten so we know that c is the center of the circle and c d is perpendicular to a b now whenever you have a segment that extends from the center of the circle and it's perpendicular to a chord then the chord is going to be bisected into two congruent parts so that means that ad is congruent to db which means d is the midpoint of chord a b so what can we do with this information well first we need to calculate the value of x let's say if a d is five db is five and a b has to be ten so therefore a d is one half of a b five is one half of ten so ad is one half of a b now a d is 3x minus 1 and a b is 7x minus 5. let's multiply both sides by 2. so one half times two is one so we're just going to have seven x minus five on the right side on the left it's going to be six x minus two now let's subtract both sides by six x and let's add five to both sides negative two plus five is three seven minus six is one so x is equal to three now that we have the value of x we can calculate a d now a d is three x minus one and so it's going to be three times three minus one three times three is nine nine minus one is eight so a d is eight which means db is also eight and a b has to be 16. now let's draw our line from the center to point a so anytime you draw a line segment between the center of the circle and any point on a circle that represents the radius of the circle so if ac is the radius of the circle then ac is 10. and notice what type of triangle we have this is related to the 3 4 5 triangle if we multiply by 2 this will give us the 6 8 10 triangle so the missing side is 6 which is cd and so that's how we can calculate the length of segment cd it's equal to 6 which means that b is the answer number 31 the arithmetic mean and geometric mean of two numbers are 35 and 28 respectively what are the two numbers so first let's talk about the arithmetic mean and geometric mean for example what exactly is it the arithmetic mean is related to an arithmetic sequence so for example 4 7 10 13 16 forms an arithmetic sequence there's a common difference of three between each numbers so if i want to find the arithmetic mean of 4 and 10 it's going to be the middle number of the sequence and the formula to calculate the arithmetic mean it's going to be a plus b divided by 2. so it's going to be 4 plus 10 over 2. basically it's the average so 14 over 2 is 7. now let's say if i want to find the arithmetic mean between 4 and 16 it's going to give me the middle number 10. so using the same formula 4 plus 16 divided by 2 that's equal to 20 over 2 which is 10. so that's the basic idea behind the arithmetic mean now as for the geometric mean it's very similar but let's write a geometric sequence so let's start with four and i'm going to multiply by two eight sixteen thirty-two sixty-four so to get to the next number you need to multiply by two so that's the geometric sequence we have a common ratio of two so the geometric mean between four and sixteen is eight and here's the formula to calculate it's the square root of a b so the square root of 4 times 16. now 4 times 16 is 64 and the square root of 64 is 8. now let's say if i want to find the geometric mean between 4 and 64. it's going to be 16. now what's 4 times 64 4 times 60 is 240 4 times 4 is 16 so 240 plus 16 is 256 and the square root of 256 is 16. now granted you could have done it this way you could separate 4 and 64. the square root of 4 is 2 the square root of 64 is 8. 2 times 8 is 16. and this shouldn't be square root 16. that's simply 16. so as you can see the geometric mean is 16. so now you understand what the geometric mean and a different sigmen represent let's focus on answering this problem so the arithmetic mean which i'm going to call a is 35 the geometric mean is 28. what are the two numbers so the arithmetic mean is going to be the two numbers a plus b divided by 2 and the geometric mean is going to be the square root of the product of those two numbers so we have two equations and two variables how can we solve for these two variables well let's focus on this equation i'm going to isolate a so let's multiply both sides by 2. 35 times 2 is 70. so a plus b is equal to 70. subtracting both sides by b i have that 70 minus b is equal to a now on the right side i'm going to get rid of the square root by taking the square of both sides so 28 squared or 28 times 28 that's 784 and these two will cancel so i'm just going to have a times b now i'm going to replace a with 70 minus b so now i could solve for b so i'm going to distribute this b so it's going to be 70 b minus b squared so i'm going to take everything from the right side and move it to the left side so on the left is going to be positive b squared and negative 70 b now this is going to remain on the left so it's going to stay positive 784. now i need to factor the expression so i need to find two numbers that multiply to positive 784 but add to negative 70. so let's start with four 784 divided by four is 196. that's not going to add to seven uh that's not going to add to negative 70. now let's try 8. 784 divided by 8 is 98 so that's not going to work let's try 16. this is going to be 49. 49 plus 16 is 65 so that's close now 49 contains two sevenths so i know that 784 is divisible by 7 and 16 can be broken into 8 and 2 4 and 2 and 2 and 2. so i need to try a number that's less than 16 so i'm going to try 7 times 2 which is 14. 784 divided by 14 is 56. notice that 14 plus 56 adds up to 70. but i need to make it negative so to factor this expression it's going to be b minus 14 times b minus 56. so now we have two possible answers for b so let's set each factor equal to zero so b can be equal to 14 or it could be equal to 56. i recall that earlier we said that a plus b is equal to 70. so if b is 14 70 minus 14 is 56 and if b is 56 a is 14. so either way you see it the two numbers are 14 and 56 and so that's the answer number 32 calculate the lengths of segments ac cb and cd in the figure below so what we have in this problem is an altitude on the hypotenuse so we have altitude cd and it's on hypotenuse a b of triangle acb whenever you have a triangle like this there are some equations that you want to be familiar with but first let's label the variables that we need to be familiar with so ac i'm going to call it lowercase b because it's opposite to angle b c b i'm going to call it lowercase a it's opposite to angle a and the altitude cd i'm going to call it h it's the height of the triangle that is triangle acb so we need to calculate a b and h in this problem now i'm going to call 12 or adx db is y and the sum of x and y is c because that's opposite to angle c now to calculate h you need to know that h is the geometric mean of x and y a is the geometric mean of y and c and b is the geometric mean of x and c since x is closest to b so those are the formulas that you need to know to calculate a b and h so let's start with h x is 12 and y is 6. so it's the square root of 72. now 72 is 36 times 2 and the square root of 36 is 6. so h which is segment cd that's 6 square root 2. so that's one of the answers that we need now let's calculate a segment cb so a is going to be the square root of y times c so y is 6 and c is 12 plus 6 or 18. now 18 is 6 times 3 and the square root of 6 times the square root of 6 is 6. so a which is segment cb that's 6 square root 3. now let's move on to the last part and that's b so x is 12 and c is 18. so 12 is 6 times 2 18 is 6 times 3. and i'm going to multiply two and three so there's three sixes in this example six times six is six so b is going to be six square root six and so that's segment ac and so that's it for this video so make sure you know that these three formulas are basically the geometric means of these values now there are some other equations that are useful if you ever need them so if you focus on the large triangle that is triangle acb it's a right triangle so therefore you can use the pythagorean theorem so you could say that c squared is equal to a squared plus b squared so that's another form that you could use if you need to find a missing side now notice that we have a right triangle here as well between b h and x so therefore we could say that b squared is equal to h squared plus x squared and there's also another right triangle here so we can also say that a squared is h squared plus y squared and also we know that c is the sum of x and y as you can see so those are seven formulas that you can use for this particular situation 33 what is the area of the parallelogram shown below now this equation or rather this problem is not that bad you simply need to know this equation the area of a parallelogram is just the base times the height so the base is 10 the height is 15 so it's 10 times 15 which is 150 square units and so that's a simple way to calculate the area of a parallelogram 34 what is the sum of r and z in parallelogram abcd so the first thing that we need to know is that opposite sides are congruent in a parallelogram so we can say that 24 is equal to 3x plus 9. this will help us to calculate x and we can also say that 30 is equal to 4y plus 2. so let's subtract both sides by 9. 24 minus 9 that's 15. and we can bring down the 3x and now let's divide both sides by 3. so 15 divided by 3 that tells us that x is equal to 5. now in this equation let's subtract both sides by two so 30 minus two is 28 and then let's divide by four so 28 divided by four is seven so y is equal to seven so now that we have x and y we can calculate r and z the next thing you need to know is that the diagonals they bisect each other so r is equal to five x plus 2 and z is equal to 3y now we have the value of x x is 5. 5 times 5 is 25 and 25 plus 2 is 27 so r is equal to 27. now z is equal to 3y and y is 7. 3 times 7 is 21. so z is equal to 21. and now our goal is to calculate the sum of z plus r so that's 21 plus 27 which is 48 so that's the answer which is answer choice e 35 what is the difference between the values of r and s in parallelogram abcd so in a parallelogram you need to know that the opposite angles are congruent so a equals c b is equal to d now b and d won't be helpful because we have two different variables however we need to focus on a and c because they both contain the same variable x so the measure of angle a is equal to the measure of angle c so angle a has a measure of 13x and the measure of angle c is 10x plus 15. let's subtract both sides by 10x 13x minus 10x is 3x so 3x is equal to 15. and if we divide by 3 we can see that x is equal to 5. so now that we have the value of x we can determine the measure of angle a and c so the measure of angle a is 13x which is 13 times 5 and that's 65 degrees so angle a is 65 degrees and angle c which equals angle a must also be 65. now angle c and angle d consecutive angles in the parallelogram are supplementary so the measure of angle c plus the measure of angle d they add up to 180. so c is 65 and we're looking for d so d is going to be 180 minus 65 which is 115. so if angle d is 115 angle b is also 115. so now we can calculate the values of r and s so 115 which is angle b that's equal to 12r plus 19. so let's subtract both sides by 19. 115 minus 19 is 96 and that's equal to 12r 96 divided by 12 is 8. so r is equal to eight now angle d is eleven s minus six and that's also equal to one fifteen so let's add six to both sides 115 plus 6 is 121 and if we divide both sides by 11 121 divided by 11 is 11. so we have the values of s and r and our goal is to calculate the difference between those two values so s minus r that's 11 minus 8 the difference is 3 which means that c is the answer and that's it for that problem 36 what is the value of h in the figure below now we have a right triangle and in order to calculate h you need to be familiar with something called sohcahtoa the so part tells us that the sine of the angle is equal to the opposite side divided by the hypotenuse k tells us that cosine of the angle is the adjacent side over the hypotenuse and toa tangent tells us that tangent theta is going to be the opposite side over the adjacent side so the hypotenuse is the longest side of the triangle it's across the box relative to the 25 degree angle 500 is adjacent or next to it and opposite to the angle is h so therefore since tangent is opposite over json we want to use that particular trig function so tangent of the angle 25 degrees is equal to the opposite side which is h divided by the adjacent side 500 so let's cross multiply this is going to be 1 times h and that's equal to 500 times tangent of 25 degrees make sure your calculator is in degree mode so 500 times tan 25 that's equal to 233 rounded to the nearest whole number and so that's the answer 37 calculate the area of the regular hexagon shown below the area of a regular polygon is one half the apothem times the perimeter now if it's a regular hexagon that means all sides are the same so each side is 20. so the perimeter is going to be n times the length of each side so the six sides and each with side length of 20 so the perimeter 6 times 20 or 120 so now we need to calculate the apothem which is the distance between the center of the circle and the midpoint of any side of the circle so that's the apothem between the center of the circle and the vertex or any vertex that's the radius of the circle i mean not the circle but the polygon so how can we calculate a so if we have let's say a hexagon and we split it six ways what do you think the angle is for these six pieces that i drew a full circle is 360. and if you take 360 and divide it by 6 that means that each of these angles represents 60 degrees so therefore this angle 60 which means the angle inside this triangle is half of 60 which is 30 which means that is 60. so let's focus on this right triangle this 30-60-90 triangle our goal is to calculate a now this side length is 20 which means that the base of this small triangle is half of 20. which is 10. so how can we calculate a if this side is 10 now you need to be familiar with the 30 60 90 reference triangle across the 90s 2 across the 30 is one half of the hypotenuse and across the 60 is whatever this number is times the square root of 3. so if this is 10 this number is going to be 10 times the square root of 3. and so that tells us the value of a so now that we have a and we also have the perimeter we can now calculate the area using this formula so it's going to be one half multiplied by the apothem which is 10 square root 3 times the perimeter which is 120. now 10 times 120 that's 1200 and half of 1200 is six hundred so the area is going to be six hundred times the square root of three and that's the answer thirty-eight calculate the total surface area of the figure shown below so what we have here is a triangular prism and we have a right triangle so the hypotenuse has to be five it's a three four five right triangle now the total surface area of a prism or triangular prism it's going to be the area of the base plus the lateral area so the area of the base represents the triangles the triangle on the left and the one on the right now the area of a triangle is base times height well one half base times height but there's two triangles so we're going to multiply that by two the lateral area is the perimeter of the triangle times the height of the prism so we could cancel two and one half so the total surface area is going to be the base of the triangle times the height of the triangle times the perimeter of the triangle multiplied by the height of the prism which is 10. so the base of the triangle is 3 the height of the triangle is four the perimeter of the triangle is going to be three plus four plus five which is twelve and the height of the prism that's ten three times four is twelve twelve times 10 is 120 and so 12 plus 120 is 132. so the total surface area is 132 square units 39 what is the measure of arc ef now we're given angle b so that's 40 degrees and arc df is 120. how can we use this information to calculate the measure of arc ef now let's say if you have a circle and you have a point and let's draw two tangent segments that extend to that point so let's call this a b and c so b is the common endpoint of two tangent segments and keep in mind a tangent or tangent segment touches the circle at only one point you need to know that these two tangents are congruent and also angle b let's say if it's 50 and arc ac which is going to be 130 these two are supplementary so knowing that if angle b is 40 arc d e has to be 180 minus 40 which is 140. now if arc df is 120 angle a has to be 180 minus 120 which is 60. now if b is 40 and a is 60 we know that c has to be 180 minus 60 minus 40. 180 minus 60 is 120 120 minus 40 is 80. so angle c is 80 degrees now if angle c is 80 arc ef has to be 180 minus 80 which is 100 now we can check our answers so de is 140 df is 120 and ef is 100. these three have to add up to 360. 120 plus 140 plus 100 does add up to 360 which is a full circle so that's how you can quickly check the work to see if you did it correctly so the measure of arc ef is 100 degrees and that's the answer for this problem 40 what is the sum of the radii of the three circles shown below so we're given a b a b is 13 and bc is 12 and we're given also ac which is 15. so how can we find the answer that we need now notice that this is the radius of circle a but first let's call this point point d this point here point e and point f now these two are equal to each other because that represents the radius of circle a these two parts are congruent that represents the radius of circle b and these two parts are congruent which represents the radius of circle c so let's call the radius of circle a x so this is x as well now if a d is x what's b d b d has to be the total length minus x or 13 minus x now fc has to be 15 minus x because when you add these two you need to get 15. now because db and be are the same be is also 13 minus x and fc and ce are also equal to each other so c e is 15 minus x now we could say that c ce which is this segment plus be which is that segment that's equal to the total segment cb or bc now ce is 15 minus x be is 13 minus x and bc is 12 so now we can calculate the value of x so let's combine like terms 15 plus 13 is 28 and negative x minus x is negative 2 x so let's subtract both sides by 28 12 minus 28 is negative 16 and now let's divide both sides by negative two so x is equal to eight so now we can calculate the radius of each circle so let's start with the radius of circle a notice that it's equal to x and x is eight so r a is equal to eight now for circle b the radius is thirteen minus x so this is going to be thirteen minus eight which is five now let's move on to circle c for circle c it's 15 minus x which is 15 minus 8 and that's 7. so now we have the radius of all three circles so to calculate the sum it's going to be eight plus seven plus five now eight plus seven is fifteen fifteen plus five is twenty so the sum of the radii of all the circles is 20 units and that's it so that's it for this video thanks for watching and if you want more example problems on geometry like the ones you've seen in this video just check out my geometry playlist and you could find a specific topic that you need help with thanks again for watching you