hello grade 11 students your favorite maths teacher is here today with another popular mathematician who's that Pythagoras Oh Pythagoras the correct term is Pythagoras because one of my Greek student told the correct pronunciation is Pythagoras but British people use Pythagoras we are influenced in British way so I'll call Pythagoras theorem so every day you need Pythagoras theorem even the carpenter needs Pythagoras theorem for making chess so he needs 90-degree angles Mason's look at 90-degree angles in foundations walls so without knowing everyday we are using Pythagoras theorem so basically Pythagoras theorem is related to right angle triangles so what is right triangle triangle means one angle should be a 90-degree angle so we'll see how we can apply Pythagoras theorem in real-life applications now look at this triangle ABC triangle so here marked right angle so before we start Pythagoras theorem we need to identify the sides so we call opposite to ninety degree angle the side AC is the hypotenuse and the other two sides we just call two sides hypotenuse you have to identify the other two sides so what is hypotenuse here a see other two sides a B and BC what's the right angle B angle is the right angle look at this triangle PQR triangle which angle is 90 degrees P angle is 90 degrees so what's the hypotenuse opposite to right angle Q R is the hypotenuse then what are the other two sides other two sides P R and P Q look at this triangle it's also right angled triangle which angle is 90 M angle is 90 then what's the hypotenuse KL is the hypotenuse then what are the other two sides other sides are K m and M look at carefully without the diagram you can write down the other two sides and their hypotenuse now look at B angle here for the hypotenuse B letter is not there for the two sides starting from here a B BC look at this one P angle is 90 then what's the other side QR without the P letter then what are the other two sides Q P or P Q and P are you can connect with the hypotenuse M is 90 degrees then without em later the other two letters comes in the hypotenuse and further other sides M comes now we'll see without the diagram write down hypotenuse what's the hypotenuse here without white letter X Z what are the other two sides X Y and Y Z you have you can connect violator with the hypotenuse DF is a triangle D angle is 90 so what's the hypotenuse their hypotenuse is without D yeah then what what are the other two sides the two sides are in D and D so you should be able to identify the 90-degree angle what's the hypotenuse L are the two sides now again you need to know how to expand brackets so we'll take some examples and see whether you can remember these a plus B whole things square so what's the easiest way to do square the first double of first and last two times a and B multiplication plus 2 a B and then last term squared you get three terms don't forget that because most of the students I have seen just writing a squared plus B squared that's wrong you are getting middle term as well what's this one square of the first term square of the last term comes here always positive then the middle term becomes the two times the product of these two minus 2xy this one two sides BC and CD you can do the same thing BC squared C d squared comes over there then what's the middle term double of these two plus two B C and C DS now try this one what's the faster a B squared what's the last term B C squared what's the middle term - doubled a B in to PC now let's take this one a B is given C D over 2 when you screw it what happens when you screw it you need to square the side as well the whole thing square so what is that you can write down CD squared over 4 2 squared becomes 4 if X y equals a B over 2 what's X Y squared when you square both sides you get a B squared over 4 2 into 2 4 so remember when you have an expression when you want get the square term square both sides now take this one PQ is 2 CD when you want P Q square have you do to CD whole thing square so 2 times 2 4 C D Square when it's 3 CD same thing you need to square the whole thing when you square the whole thing three times three comes three times three is nine C D Square this one square both sides what happens here you have to square this term as well as this one so you get 2 into 2 4 4 XY squared is equal to 3 into 3 9 ml square what about this one 5x y equals 3 P Q square both sides you get 25 XY squared and 9pq square so you need this one when you apply Pythagoras theorem now look at these diagrams and find area this is a right angle triangle what's area 1/2 times base times perpendicular height 1/2 B 1/2 a B this one two sides X&Y area is 1/2 times base is X perpendicular height is y 1/2 X Y this is a square what's the area of a square area of a square is X into X or you can write X square because the side is given as X what's the other shape trapezium how do you find out the area of the trapezium add the two parallel lines it must be divided by two and multiply by the perpendicular height H what about this one this is also a trapezium these are the two parallel sides so what's the area area equals 1/2 what are the two sides B plus C is the sum of the parallel lives what's the perpendicular height B plus C B plus C B into B plus C you can write B plus C square look at this one and we'll try to find out when you take a B what's the square on a B side so if you draw a square on a B side you get same length as this so what's the square area of the square on a b-side area of the square on a b-side you get a B into a B a B squared what's the area of the square on BC side area of the square on BC side what's area so when you have to you have to draw a square here BC into PC so that's BC square likewise what's if I draw a square on AC side what's this area area of the square on AC side I get a squared AC into AC again we will take the triangle if I want the a square on a B side what's the square on a B side the area of the square on a B side that's a B squared then what's the area on this side area of the square area of the square on be seized then that becomes you get it square here we see into BC BC square then what's the area of the square on AC side that's AC square so you need to know the area of a square on all sides now we'll try to see what's Pythagoras theorem is now for that I'm doing a small activity so we'll take a triangle with side length three and four I take side length three here one two three four four and three so what's the child here I get this child for squares here three squares up then I want to draw I want to get the square on all sides so I'm taking this one as a B C this is 90 degree and so this is four squares three squares so what's the area on BC side I'm getting one two three four so here I'm getting a square 4 by 4 and here what's the area on a B this is three lengths so I'm getting three by three square I'm getting three by three square so here I'll try to color that this is 3 by 3 and this is 4 by 4 so what's available so this is nine I'm getting nine this is how many squares I'm getting 4x4 16 as the area now it's difficult to draw a square there we can't measure properly so what I do I am doing this trick to draw the square on a seaside so I'm taking this is 4 so I'm taking three squares here barn I'm taking three squares here 1 2 3 & 4 1 2 3 4 draw this line here what I do here 3 there I'm going 4 squares up I'm going 4 squares up 1 2 3 4 & 3 squares here then connect here to here again I'm doing 4 squares to the right and 3 squares down here to here then I connect I want to see what's the area is I can show you this is a square with one side AC how can we show that now take this is 90 degrees and this is a this is alpha and this is beta so 90 alpha beta total is 180 so this is 90 s they are so alpha plus beta is 90 degrees so if this is a 4 now look at this is 4 & 3 triangle so 3 & 4 same trial other way around so what is alpha so this is 90 degrees so what is alpha alpha now along this side four four sides so alpha is here then definitely this should be beat here this triangle also same thing three by four three side and four side whereas alpha alpha is here beta is here and this is 90 degrees this way also same same angle so this is alpha this is beta and this is night now look at this one also beta is there on this point so what should be this angle this is a straight angle this should be 90 degrees here alpha beta is they are so this should be 90 degrees half a beta is there this is 90 degrees alpha beta is 90 degree so I drew a square with AC such now I want to check what's the area of this one to do that I can take the whole picture this whole square and get the area subtract all the four areas of triangles so I'll do that and see whether I'm getting a certain answer related to this one okay we'll do it how many squares this way one two three four five six seven seven by seven square so what's the area seven by seven 49 then what's the area of this one one triangle is half times one length is four other one is three so two times two two times three six one triangle is 6 so this is 6 this is 6 6 6 so 6 times 4 24 then the whole thing is 49 49 minus 24 what's there we are here 49 minus 24 you get 25 here now do you get any relationship between these three numbers this is 9 16 25 what do you think 9 plus 16 is 25 so this Plus this is equal to this so we get a relationship 9 plus 16 is equal to 25 so Pythagoras found the relationship like this when you take a triangle right-angled triangle these two sides are these two and this is the hypotenuse so this is the square along the hypotenuse square along the hypotenuse these are the other two sides so he found out a connection between the areas you are getting with the squares so we'll see what Pythagoras mention about the theorem in a right angle triangle the area of the square drawn on the hypotenuse so if you take this triangle this is the hypotenuse I'll label ABC triangle AC is the hypotenuse so in the right angle triangle the area of the square drawn on the hypotenuse this area is equal to the sum of the areas of the squares drawn on the remaining sides of the triangle which include the right angle so that's what we found here these are the two sides this is the hypotenuse so the area of the square formed on the hypotenuse is equal to the sum of areas of the squares on the other two sides that's what the Pythagoras theorem is so we'll try to prove with different ways so I'll take this diagram and try to prove the Pythagoras theorem according to Pythagoras theorem so this is B angle is 90 and this is the hypotenuse so he mentioned a squared is equal to area means a 2a what is this one B squared plus C square now we'll try to prove that using this diagram so what I did I took the other length here to this triangle I added if this is see I added be live and C length here and then this way you get hypotenuse there so what is the full shape you are getting a b c d e so what is a BCE diagram a BCE is a trapezium you are getting in the trapezium we will try to find out areas separately and add it to get the area of the trapezium so what's the ABC triangle area half B C plus what's this triangle AC e triangle half a times E what's CDE triangle half B into C so what's that half BC half BC you get b c plus half a square now if you are taking the trapezium the whole trapezium these two sides are parallel so what's the area of the trapezium when you use the formula straight away a BCE trapezium B plus C divided by 2 B plus C divided by 2 and then multiplied by the perpendicular length B plus C B plus C into B plus C I can write B plus C whole thing squared over 2 now expand brackets what's easiest way to expand square the first term square the last term and you are getting the middle term as double of these 2 BC so what you are getting here when you divide you get B square plus C square divided by 2 plus 2 to get canceled out BC now we know these two are equal so this is first one this is second 1 1 & 2 equal so I can write BC plus half a squared is equal to B squared plus C squared over 2 plus BC BC BC get canceled out multiplied by 2 the whole equation multiplied by 2 you get a squared is equal to B squared plus C squared so what we need here so that's the Pythagoras theorem now we'll take another diagram and see now this time this is the ABC triangle so here what you need to prove a squared is equal to B squared plus C squared now I'm taking if this is C this way be other way around c b:c be I created ur square like this so what's the area of the whole thing this is B plus C B plus C so the area of the whole square is B plus C squared B plus C 2 B plus C so when you expand that I get B squared C squared and the middle term to be C if I take separately for triangles and the square what's area I can do the same thing area of square what's the Triangle area half BC half BC half BC how many triangles you get one two three four triangles plus area of the square ha into a you get a square so when you simplify you get 2 BC plus a square now both are equal equate 1 and 2 but you get one equals two you get P squared plus 2 BC plus C squared is equal to 2 BC plus a squared 2 bc 2 BC get canceled out so what you get B squared plus C squared is is color so same thing there so this is another way of showing the Pythagoras theorem now we'll take the form of proof so take the ABC triangle and data is we know this angle is 90 and we need to prove that this is the hypotenuse be B C squared is equal to a B squared plus P Sisco AC square we are doing a construction here ad is a perpendicular line now we'll try to do the proof proof we'll take a DC triangle ADC triangle and ABC triangle what can you say about those two triangles ABC I'll put alpha here and beta here so alpha plus beta plus 90 is 180 so same thing here this is 90 so alpha plus beta should be 90 because 90 be R so if this is beta what should be this angle this is alpha if this is alpha this is beta so we'll try to find out the angles in ATC angle we know D angle is equal to a angle that's 90 degree angle giver then if you take this angle C angle that's alpha angle is equal to C angle in the other triangle beta angle is a angle in the ABC triangle what's that beta equals to B angle in the big triangle that's beta so we know these three angles are equal so these two triangles are equi angular so if there is two triangles are equal and Euler these are similar triangles so we can write down the corresponding ratios so the corresponding ratios are equal so corresponding ratios will take opposite to 90-degree angle BC then opposite to 90-degree angle is a see ya is equal to we'll take opposite to be it aside that's C D opposite to beta side for the bigger one that's AC and here beta side for this one is C D so this is my first equation then I'm taking other two triangles the other one and the big one abd triangle and abd triangle any BC triangle so similarly I can show that these two are equal because a angle is 90 degrees that's equal to the angle in this triangle then alpha angle here that's a angle and this is C anger that's alpha B triangle is B angle for ABC angle and the other angle also be end so here also equiangular triangles then these are similar triangles so the corresponding ratios are equal what are the corresponding sides there if you take opposite to 90 that's BC opposite to 90 here that's a be opposite to alpha that's BD in the smaller bar opposite to alpha that's a B now we'll take this one and this one has to look at the two ratios from the first equation I want to cross multiply and write down AC into AC as AC square is equal to BC into CD from this say question also I want to cross multiply a B squared is equal to PC into BD I like these two a b squared plus AC squared because we need a b squared plus AC square when you add these two what happens a b squared plus AC square you get b c BD plus BC CD PCs comma take it out BD plus CD or DC what's BD plus TC this Plus this is BC instead of this one I can write BC so BC into BC you get see square so according to Pythagoras theorem BC squared is equal to a B square plus AC square now we'll try to write down the relationship between Pythagoras theorem with these sides this is a right angle triangle so you can apply Pythagoras theorem what's the hypotenuse a opposite to nine so we can drive square rough the hypotenuse is equal to the sum of squares of the other sides B squared plus C squared when you take this triangle how can we write down the relationship we can write AC squared is the hypotenuse squared what are the other two sides a B squared plus BC square now look at the other triangle which one is the hypotenuse opposite to 90 degrees Cuba so I can write cube R squared is equal to PQ squared plus PR square if I want the other side if I want PR how can I write P R square PR squared is P Q R squared - so always hypotenuse squared is first and then you can subtract the other side so Q R squared - you want PR so PQ if I want the other side PQ R how can I write P Q R P Q squared P Q squared is equal to Q R squared minus P R square if I want in this triangle a B squared how can I write a B squared is AC squared minus BC square now take a lemon triangle and write down what is KL square KL squared is now the hypotenuse so came squared minus LM squared so what is K M squared K M Squared is the hypotenuse so you can add KL squared plus L M square so remember if you want the hypotenuse side you can add the other two sides if you want another side without the hypotenuse you have to subtract from the square term of the hypotenuse so fine unknown lengths look at this triangle so hypotenuse a is a so I can write a squared is equal to 3 squared plus 4 squared what is 3 squared 9 4 squared is 16 16 plus 9 25 so how can we find out a square root of 25 plus or minus 5 but we take plus 5 because this is a length so 5 centimeters do this 1 B is not the hypotenuse so what can we write B squared is equal to 10 squared minus 6 square 10 squared is hundred 100 minus 36 you get 64 square root of 64 you get 8 8 centimeters so remember these measurements three four you get five here six 10 and this is 8 now look at this one C is not the hypotenuse so how can you write down C squared is equal to 15 squared 15 squared minus 12 square so what is 15 squared 225 12 square 144 so when you subtract what you get you get 169 square root of 169 is 13 so that's thirteen centimeters 12 13 and 50 this one D is the hypotenuse now so how can you write down d squared is equal to 5 squared plus 12 squared Y squared is 25 12 squared 144 you get hundreds 69 square root of 169 is 13 centimeters so 5 12 13 this one is not the hypotenuse so you can write e squared is 26 squared minus 24 square so what is 26 square 276 we'll check it 26 seven to 26 676 676 - 24 times 24 20 and 4 2 times 4 8 2 times 2 4 4 times 4 16 4 times 2 8 + 1 9 so you get 6 576 so you get 100 their square root of 100 is 10 so you get 10 centimeters for a letter in this one F so you need F squared is equal to 30 9 squared - 36 square here you can use difference between two square terms I can write 39 plus 36 39-36 it's easy when you use them difference between two squared terms 9 + 6 15 1 remaining 7 7 - 5 times 3 but 75 times 3 225 square root of 225 is 15 centimeters so square root of 225 is 15 centimeters so this is 15 centimeters so look at these numbers we got 3 4 5 6 8 10 12 13 15 5 12 13 we'll find out what are these we call that these are Pythagorean triplets so if we found out three four five is a triplet three for one side and the hypotenuse is 5 then we found out here can you remember from the previous page 5 12 13 5 12 13 so we can find out any triplet this way as well as this way so here 3 4 5 and then we found out 6 8 10 so what's that 6 8 10 multiply this by 2 what's the other one 10 24 26 we got that one so here when you multiplied this by two you can get 680 x 3 3 x 3 9 3 x 4 12 3 times 5 15 that's also a triplet so this way you can find out many as you like multiples of any number you get a triplet here also multiplied by 2 10 24 26 that's a multiple so here my you can multiply by 3 4 or 5 or you can divide by 2 this one when you divide by 2 you get 1 point 5 2 & 2 point 5 that's also triplet so any multiple you get a triplet but how can you find out this way what's the pattern there observe carefully 3 is there hi what do you think the next number is 3 5 7 how can we find out the other 2 numbers look at carefully there's a pattern here what I can do take the number it's quite 3 squared 9 subtract 1 8 divide by 2 you get 4 and the next number is always here adding 1 to that then the next number is 5 look at the same thing here 5 squared 25 subtract 1 24 divided by 2 you get 12 when you add 1 you get 12 and 30 then what are these 2 numbers 7 squared 49 subtract wall forty-eight divided by two forty eight divided by two you'll get 24 so what's the next number 25 so seven 24 25 also a Pythagorean triplet so that's how we can find out Pythagorean triplets this way you can take any multiple this way what you do Square the first subtract 1/2 now we'll see how we can find out the triplets if you have given one value 9 is given you can take 3 out when you take 3 out this is 3 but we know if this is 3 3 squared is 9 9 minus 1 8 8 divided by 2 4 4 & 5 are the triplets so what's the triplet now you have to multiply by 3 so 3 times 3 you get 9 so what's the other two numbers 12 and 15 look at 12 I can take Fallout then if this is three what are the other two numbers triplets what are the triplets three squared is nine nine minus one eight eight divided by two four four five then if it's twelve what are the other two numbers 16 and 20 so here remember that always try to keep the odd number inside then it's easy to because you're subtracting one and find the other number so here three five seven dollar odd numbers so here keep the first number as odd number 15 what's the number I can take I can take three out and keep five oh I can't take other way round as well because both are both odd number is five and keep three now what are the other two triplets if I take 5 here 5 squared 25 25 minus one is 24 24 divided by two you get 12 and the next number is 30 if you take three what are their triplets 3 squared 9 9 minus one eight 8/2 for l5 here when you multiply by 3 you get 15 36 and 39 here another triplet you are getting 15 3 times 4 12 5 times 5 20 25 so here when you take the different about you get different triplets now you know how to get all triplets so make sure keep odd number here to use this method take example number one in the right angle triangle ABC B is ninety a B's five B's is twelve find the length of the side AC so we sketch the diagram a b c a b 5b c 12 and you're asked to find AC and this is right angle so you can use Pythagoras theorem AC squared is equal to a B squared plus BC square then substitute the values v squared plus 12 squared 5 squared 25 12 squared 144 when you add you get 169 square root of 169 is plus 13 so the length of AC is thirteen centimeters example number two find the length of CD based on the information given in the figure so you ask to find out this length when you take that triangle these two sides unknown so first you have to apply Pythagoras theorem for ABC triangle and then find bc then again apply for BCD triangle so twice you have to use Pythagoras theorem so what is BC squared B C squared is 9 squared plus 12 square that's a C squared plus a B square 9 squared is 81 12 squared is 144 when you add do you get 225 square root of 225 we know that that's tea now apply the next one for the next triangle BCD triangles so BCD triangle what is C d square C D squared is B D squared minus BC square b d squared we know that 17 squared here BC square I can use 15 square use difference between two squared terms it's easy to otherwise you need to know what is 17 squared 15 squared I do 17 plus 15 17 minus 15 you get 32 into 264 square root of 64 you get the length of CD so CD equals square root of 64 that's 8 so this is in centimeters so the CD length is 8 centimeters example 3 this practical application so you can use Pythagoras theorem in real-life situations one end of a wire is tied to a ring fastened at a point 1 meter below the top of a vertical utility pole will take this the pole this is the ground so warn me to below top of a vertical utt pole while the other end is tied to a ring fastened on the ground here are something like this 8 meter away from the foot of the pole here to here so 8 meters away from the foot of the four the length of the wire between the two rings is 10 meters so this is given 10 meter away find the height of the utility pole assume that the wire is stretched so in the asking find this length first I'll put X here and you have to add one to get the length of the pole ground and the pole this is 90 degrees so we can see this is a right triangle triangle with lengths you can use Pythagoras theorem so what is x squared x squared is 10 squared minus 8 squared 10 squared is 100 8 squared is 64 hundred minus 64 you get 36 when you take the square root that's 6 centimeters so this is 6 so what's the height of the whole height of the pole is 6 plus 1 7 meters because this is given in meters exercise 17.1 fill in the blanks using the information in the figure M o squared this is hypotenuse how can we write M n squared plus a no squared B d squared BD is this one we can take the abd triangle so what is BD squared a B squared plus BD is this one B a B square plus ad square then AC squared plus CD squared that's a CD triangle that's a d square so what is a B squared look at these two ad square ad squared get cancel out you get a B squared is equal to without that what is a B Square so what is a B squared now take this triangle ABC triangle so what's that AC square plus BC square look at this one PQ square where's P Q P Q we can use this triangle so P Q squared is P T squared plus T Q squared Q R square Q R squared this one so this is also 90 degree so how can we write Q R squared I can write Q T squared plus T R square this one a B Square a B Square I can use the ABC triangle AC squared plus BC square then a squared plus EC square a e squared plus C squared is AC squared then ad square ad is this one so that's to the ACD triangle you can write AC squared plus CD square so you need to identify the triangle and then apply the Pythagoras theorem question number two find the value of x in each of the right angle triangles given below X is hypotenuse so x squared is sum of squares of the other two sides 36 Plus 64 you get 100 square root of hundred you get that's 10 centimetres look at this one it's a right triangle triangle you can straight to be applied Pythagoras theorem x squared is equal to 13 squared minus 2 squared 13 squared is 169 12 squared 144 when you subtract you get 25 square root of 25 is 5 now if you are not doing with square terms how can we use difference between two square terms I can do 13 plus 12 and 13 minus 12 so what is 13 plus 12 25 times 1 that's 25 so sometimes it's easy to do that way part 3 X is here first we'll apply to this triangle and find out km squared km squared is root 5 squared minus root 2 square root 8 root Phi 2 root 5 you get 5 root 2 into root 2 is 2 5 minus 2 is 3 so km squared is 3 then apply to km in triangle km and triangle you can apply Pythagoras theorem x squared is equal to km squared plus 1 squared km squared is 3 3 plus 1 4 square root of 4 is X 2 centimeter square root of 4 I can take as to this one X is here but first we need to find out PR square first find PR squared that's 4 square plus 3 squared 4 squared is 16 3 squared 9 you get 25 now use PQR triangle and apply Pythagoras theorem again x squared is equal to 12 squared plus P R squared PR squared we found that's 25 2012 squared is 144 plus 25 you get 169 square root of 169 you get 13 centimetres 4x question number 3 in the equilateral triangle ABC D is the foot of the perpendicular drawn from the vertex 8 if the length of a side of a triangle is 2 centimeter all these 2 centimeters so what happens when you draw a perpendicular line this equally divided 1 centimeter here and 1 centimeter here find the length of AD express the answer as a search now we can apply Pythagoras to abd triangle we need ad squared so what is ad squared ad squared is equal to 2 squared minus 1 squared 2 squared is 4 1 squared is 1 4 minus 1 is 3 so what's 80 square root of 3 so scrap it's a third value we can write like this question number four the location Q is reached from the location P on the horizontal ground by traveling 15 meters to the north from P and then 8 meters to the piste so normally we take not up so there's a position there our location P on the horizontal ground so if this is the horizontal ground P is here the location Q is reached from the location P on the horizontal ground by traveling 15 meters not and eight meters east we know that not an East that's perpendicular eight meters not draw the sketch based on the above information so this is Q find the distance PQ so this is right angle triangle so we can use Pythagoras theorem BQ squared is equal to 15 squared plus h squared 15 squared plus h squared 15 squared is 225 plus 64 you get 289 square root of 289 is what's the value 17 289 is 17 17 times 70 you get 289 we'll see so you get 289 so square root of 289 is 17 centimeter so PQ length is 17 not centimeters what's the length meters because it's given in meters 17 meters question number 5 the length of the diagonals of a rhombus are 12 and 60 rhombus means all sides are equal the diagonals bisect each other at 90 degree as well diagonals bisect at 90 degree so here I can write mine is 12 and the other one is 16 so 12 means 6 and 6 here 16 means a 10-8 find the length of each side of the rhombus so rhombus I'll take this one as X this is a right angle triangle I can use Pythagoras theorem x squared is equal to 6 squared plus 8 squared 6 squared is 36 8 squared is 64 you get 100 square root of 100 is 10 centimeters so what's the length of one side 10 meter so here ten so what's the side of the rhombus ten meters question number six the figure illustrates a special creation Archimedes spiral by considering the right angle triangles in the figure find the lengths ABCD efg do so here Archimedes spiral so we have to start from here so when you start from this side how can we find out if a squared is 1 squared plus 1 squared 1 plus 1 you get 2 so what is a square root of 2 so that's the length of a then you have to start from here to get B so that's why it's like a spiral it's all connected starting from here you can find all the lens there then Pythagoras theorem for the other triangle B squared is equal to a squared plus 1 squared a squared we found out that's 2 2 plus 1 you get 3 B is square root of 3 now take the next triangle C squared is equal to B squared plus 1 squared B squared is 3 3 plus 1 you get 4 so what's C square root of 4 C equals 2 now the next one C squared d squared is C squared plus 1 squared C squared we found that's 4 4 plus 1 that's 5 square root of 5 is the T again do for the next triangle here can you see we can find out all length starting from 1 root 1 root 2 root 3 root 4 root PI root like that we can find out using the Archimedes spiral then next triangle a square t square plus 1 square d squared is 5 5 plus 1 6 so E is root 6 centimeters then next one a squared so f square is e squared plus 1 squared e squared is 6 6 plus 1 7 so f is root 7 then the last one here G squared is x squared plus 1 squared you get 7 plus 1 8 so GE square root of 8 example number 1 ABCD is a square prove that AC square is equal to 2ei be square in a square all sides are equal so what can we write as AC squared if you use a BC triangle AC squared is equal to a B squared plus BC square so here a B squared B C squared so BC square is same as maybe I can substitute a B squared plus a B squared again because a B is same as B C then a B squared a B squared I'm getting to a B squared so AC squared is equal to 2a B Square example number two in the rhombus ABCD a b c d diagonals AC and BD intersects at O and also we know this is 90 degrees diagonals bisect at 90 degrees prove that AC squared AC squared plus BD square is equal to 4a B Square so we need to write out in terms of a B another important thing in rhombus diagonals bisect so here this length is equal to this length this one is equal to this length now we can apply Pythagoras theorem for this triangle a OB triangle a OB triangle what is a o squared a squared is a B squared - Oh B squared now what is a o AO is half of AC haha ac what's a B we can write a B as it is what is OB OB is half of PT you can write down why I can write this one Oh a is equal to OC as well as b au is equal to AU d now square this you get AC squared over 4a B squared half into half again one-fourth BD squared now take this one to this side I'm getting AC squared over 4 plus BD squared over 4 is a B Square so same denominator I can write as AC squared plus B T squared divided by 4 is a B squared they multiply this by 4 you get AC squared plus BD square is equal to 4 EB square example number three in the triangle ABC B is C angle BAC angle is an obtuse angle a X is drawn from a perpendicular to BC prove that a B Square - AC squared is equal to BX squared minus CX squared so we'll first find out a B squared what is a B squared a B squared is BX squared plus XA square what is AC square AC squared is a X square plus XC square now I want to subtract these two one - two you get a B squared minus AC squared is equal to be x squared plus XA or ax same thing I'll write XA here - so I'm subtracting so this becomes minus as well so X a and plus a and - get canceled out you are getting be x squared minus CX o XC same so I'll write CX because it's given as CX square exercise 17.2 ad is perpendicular to BC in the triangle ABC see figure if ad equals DC this net is equal to this length prove that a B squared is equal to BD square plus DZ squared so we'll first take a BD triangle what's a B squared a B squared is equal to BD squared plus 80 square then I can write ad square as this one using this one ad squared is same as C d square because a D is equal to TC so here what happens be d squared C D DC same thing all right TC square so a B squared is equal to TC squared a B squared is equal to BD squared plus TC square question number two a D is a perpendicular to BC in the triangle ABC ABC ad is the perpendicular to BC prove that a b squared plus c d squared is equal to AC squared plus BD square so we'll take a b squared first a BD triangle a B squared is equal to a d square plus BD square then what is a d squared according to the other triangle a DC triangle ad squared is equal to AC squared minus CD square now I want to plug this one here so a B squared ad squared is AC squared minus C T squared plus BD square then what happens I can take CD to this side I'm getting a B squared plus CD squared is equal to AC square plus vd question number three ad is perpendicular to BC this is an equilateral triangle ABC and a G is perpendicular to BC prove that for ad squared is equal to 3 BC square so all sides are equal this is also equal so we will first write down ad squared what is ad squared using abd triangle ad squared is a B squared minus BD square here I can use a B is same as BC so a B is same as BC so I can plug instead of a B this is BC square what is B D we know that this is an equilateral triangle when you draw a perpendicular line this length is equal to this then so how can I write BD half of BC now I'm getting BC squared minus half into half you get 1/4 BC square so this one I'm getting 404 BC squared minus 1 over 4 BC square when you subtract you get 4 minus 1 three-fourths BC squared is ad squared x 4 x 4 you get 4 ad squared is equal to 3 BC square question number four ad is perpendicular to BC it's marked in the equilateral triangle ABC again equilateral triangle so this length is equal to this length BC has been produced to e such that DC is equal to DC is equal to C so this length also equal proved that a a squared is 7 e C square seven of them so we'll start from ad squared we'll take a de triangle what is a e squared y squared is C squared plus d e square what is de I can write d c+ c e o I can write this is double of C because these two lengths are equal instead of this one I can write de is twice of e C I'll use the same side otherwise it's complicating EC EC squared 2 EC whole thing squared instead of de so what's the value ad squared plus 4 e C square that's a e square now I need to get rid of this ad squared so I'm taking a BD triangle when I take a B triangle what is a d square a d squared is equal to a B squared minus BD square so a B squared how can I write a B squared is PE squared B C squared because a B and VC these two names are equal that's an equilateral triangle instead of a B I can write B C squared BD BD I can write here BD length is same as c c/e squared so BD equals C then what is bc pc length I can write twice of C or E C so instead of BC I can write 2 EC squared minus c e squared or EC squared - x - 4 4 ec squared minus EC squared you get 4 minus 1 3 e C square now I can plug it here so I take this one as first one I take the first equation a e squared is equal to instead of a T squared I can use 3 ec squared plus 4 e C square so 4 & 3 you get 7 ec squared so you get Y squared is equal to 7 EC square question number 5 the diagonals of the quadrilateral ABCD bisect each other particularly at all the diagonals of the quadrilateral ABCD we'll take any quadrilateral ABCD diagonals bisect each other means this is 90 degrees at all prove that a b squared plus c d squared is equal to ad squared plus BC square so when I take a OB triangle I can write down a B squared is equal to a squared plus B squared then I need C d squared CD CD is in this triangle OCD triangle what is C d squared or d square plus C square I'll add these 2 1 & 2 1 plus 2 I am getting a b squared plus c d squared these two and I'm getting a or squared + OB squared + OD squared + OC square now look at carefully a o + OC this squared plus this squared we get AC square I take these two together and these two together a o squared + OD square Oh B Square + OC square now look at the diagram and see a o squared + OD squared now it's the right angle triangle here I can write instead of the while as a dsquared what's Oh B squared and Oh C squared I can write this one as BC square so that's what we need a b squared plus c d squared is equal to ad squared plus BC square question number six o is a point within the rectangle ABCD it's a rectangle now so o is within the rectangle inside prove that au squared plus 0 squared plus is equal to be your square plus dou square hint draw a parallel line through o to any side of ABCD so we'll take any point o draw a parallel line to any side of a B C D so here I can draw parallel line to this one and I can connect these fights I know this is a rectangle these are 90 degrees so this is 90 degrees now we need to find out a or square a or squared C your squared all these lengths so we'll do that I'll take this one as x and y what is a or squared if I take this triangle what is a over square I can write a x squared plus o x square then what is co square see your squared I can use this triangle see your squared is Oh Y squared plus C y squared now I'll take this one as one and this one is two and also I drew a parallel line to this one so this length is equal to this length and this length is equal to this length because it's a parallel line so I can add these two one and two I'm getting AO squared plus 0 squared is equal to a x squared a x squared plus o x squared plus y squared plus C by square now what is ax I can write instead of ax B bar so be by squared and what is Oh Y squared I'll add together and instead of instead of save I I can take DX we'll see whether we want that be y plus Oh Y square C Y squared C by square plus au y squared I am getting Oh C squared so I need or J squared so on I need to put instead of see why that's DX DX square now see these to be Y squared be Y squared plus o y squared what's this Oh B squared these two Oh x squared plus DX squared you get oh d square so we want this side okay now we have to write out why we equate this C Y is equal to DX and a X we took as B so when you add you are getting a all squared plus Senor squared is equal to OB or be your same thing vo squared plus OD or do same thing to you square question number seven P is a point within triangle ABC the perpendicular is drawn from the point P to the side BC AC and a B meet the size d EF respectively prove that first one B P squared B P squared minus PC squared here we need to connect BD squared minus DC square so I'm I connected this VPN PC line so we'll take what is B P squared B P squared is I can write BD squared plus PD squares then what is PC squared from this triangle PC squared is PD squared plus DC squared this is the first one and the second one I can't subtract one and two so I'm getting VP squared minus PC squared is VD squared plus DT squared minus PD squared and minus DZ squared because I subtracted the second one now can you see PD square get cancelled out so I get V T squared minus DC squared is equal to this one that's the first part second part second part b d squared plus c squared + AF squared is equal to c d squared plus a squared plus bx squared so we'll take separately be d square b d this one b d squared is B P squared minus b d square its first equation then C Y squared C is this one so how can I write P C squared minus PE square that's my second equation what's the next one F square a F squared is this one I'll connect this one with this one then f squared I can write ap squared minus P F square that's my third equation now left hand side is all addition so I'm adding 1 + 2 + 3 and we'll see what happens I'm getting BD squared plus C e squared plus a squared that's the left-hand side of the equation now look at carefully and see how we can add these two now we'll take PD PD is this one PD and which side we can connect PD with either BP BP is not BP V 2 so we can take the other side P C squared these two what happens PC squared minus PD squared I am getting CD square now we'll take another one this one PE squared P squared is this one so I can connect with AP AP squared and PE square these two what happens AP squared minus PE squared I am getting a a squared and the other one I'll take red color what about these two BP BP and PF these two I am getting real square so I am getting CD squared a squared and we have squared question number 8 the two squares abxy and b CP q lie on the same side of the straight line ABC ABC is just one line a B X Y is a square and BC BC PQ is also another square we see P kids another square lie on the same side of the straight line ABC proved that px squared I need to connect this px squared is plus C y square C y squared is equal to three times a B squared plus BC square so we know these angles are right angles and this is also right tank so we will try to find out px squared first V X square what can I write px squared I can write Q x squared plus P Q square now PQ is not there so I can write instead of P Q that's BC square because PQ is a equal to BC then QX square I'll keep as it is then next one is C by square cy squared I can write a Y squared plus a C square what is a Y a Y is not there a Y is same as a B so a VY is CMS a B so I can write a B squared plus AC square AC squared how can I write I can write AC as a B plus pieces what is Q X if I can write in terms of this length it's easy so I can write q XS Q B B Q minus bx b cubed minus bx but I know B Q is same as BC B X is same as Amy so I'm writing in terms of BC and Amy so Q X is this so I'm taking the first one and substitute take the first one px squared is equal to instead of Q x squared I can write we see - a B whole thing squared plus BC square I'll simplify this take the square term BC square middle term is minus 2 times B C a B plus a B Square and B C squared here so simplify I am getting B C squared B C squared - B C squared minus 2 a B BC and a B Square that's my I put a equation now take the second equation also I need to simplify take the second one will simplify C Y squared is equal to a B squared Plus expand records I'm getting a B squared plus 2 a B BC plus BC square so when I simplify I'm getting 2 a B squared is 2 plus BC square plus 2 a B BC let's take this is B question now we need to add the two a plus B equation what happens a plus B I am getting P x squared plus C by square that's what we need in the left-hand side then 2 BC square + BC square I'm getting 3 BC square a B squared and this - a B I'm getting 3 a B Square minus 2 a B BC plus 2 a BBC get canceled out so I can take 3 out from this and write down BC square plus a B squared is equal to px squared plus zy square so that's what we need three times a B squared plus BC square so three times ma squared plus BC square exercise 17.3 the following triples triplets or triples are the lengths of the sides of two triangles select the triangle which is a right angle triangle and write down the corresponding Pythagorean triple so how can we do that we no longer side squared is the sum of the other square terms Mills quite and see 8 squared 64 15 squared 225 17 squared 289 now I don't see these two when you add what you get because longer side is 17 when you add 64 and 225 you get nine eight - 289 so we can see eight squared plus 15 squared is equal to 17 squared so if this one satisfied means this is a right angle triangle so we can write this is right angle triangle this one largest side is 25 normally the hypotenuse is the longest side so 25 squared is 625 what is 14 squared 14 square 196 will check 14 times 14 196 then 18 squared 18 times 18 10 times 8 you get 324 when you add these two what you get 6 plus 4 10 1 is remaining 10 to 12 1 remaining 5 so in this case 625 and 520 is not equal so we can see that it's not a right angle triangle question number 2 based on the measurements given in Figure 1 and to show that BAC is a right angle in each figure be AC so this one we have to show that it's a right triangle and here also B is this one you have to show that it's a right triangle triangle so first we'll take we need to take this triangle and find out a B Square because that's the right triangle triangle what's a B squared 12 squared plus 9 squared 12 squared is 144 9 squared is 81 you get 225 square root of 225 is 15 so we got a B as 15 now we'll see the largest side is 17 so 17 squared we found out 289 eight squared 64 and a B squared is 225 when you add these two what you get 9 8 - so 17 squared is equal to H squared plus 15 squared so therefore B AC angle is 90 degrees this one we don't know this length and this length so we have to use Pythagoras twice because this is 90 so what is BD squared B d squared a is 10 squared minus 6 squared 100 minus 36 you get 64 BD equals square root of 64 that's 8 when you look at the other triangle what is DC DC squared is 7.5 squared minus 6 squared how you find out 7.5 times 7.5 at once I taught this method before 75 times 75 we do 5 times 5 25 7 times 8 56 5625 is for 75 times 75 is it 7.5 then you get you have to keep two decimal places 56 point two five if it's four point five squared what you do 45 times 45 and then put two decimal places 5 times 5 25 for with the mix number 4 times 5 20 so 20 25 so seven point five squared is 56 point two five six times 6 is 36 when you subtract 36 what you get 20 point 25 what's 20.25 four point five square root of reverse one is four point five so DC is four point five again if you have given a decimal number this decimal number have you find out the square root take 2025 we need square root of this so twenty-five square root is five so you get point five there then square root of 20 what's the closest value 25 is square root of 25 is 5 so 20 is 4 and 5 you have to find out two numbers two consecutive numbers that you get 24 and 5 so the lowest number we take lowest number of 4 + 5 is 4 so square root of 20 point two five is four point five so DC is four point five then what is BC BC is BD + DC BD is eight DC is four point five so what's that you get twelve point five now we'll look at the big triangle ABC and see whether it works we'll take ABC triangle what is BC square 12.5 square what's the easy method 125 times 125 5 times 5 25 here 12 times 30 the next number 12 times 13 so you just add 156 so you get 156 there so you need to keep two decimal places 156.25 that's 12.5 square then a B squared is 10 squared that's hundred what is AC squared AC squared is 7.5 square again what is 7.5 times 7.5 5 times 5 25 then 7 times 8 you get 56 so 56 point 2 5 then we'll see when we add these two whether we are getting this number so these 2 when you add what you get 150 6.25 you are getting 156.25 so this one is same as this one so therefore we can see that B AC is a right angle because it satisfied the sum of these two terms to the hypotenuse square question number 3 by completing the table given below find the Pythagorean triple triples corresponding to the given pairs of values verify your answers this is another method I explain you how to get the triplets or trippers so this is another way of finding Pythagorean triples so they have taken two numbers X and Y and then screw it to squared you get 4 y squared 1 then take the difference between these two that's a 4 minus 1 3 then double of XY these two double of XY 2 times 1 - 2 times 2 4 then add these 2 x squared plus y squared you get sick 4 plus 1 5 so what are the three triples 3 for fun this one 5 + 4 5 squared you get 25 4 squared 16 25 minus 16 you get 9 2 times XY 5 times for 20 20 times 240 then x squared plus y squared 5 plus 60 you get 41 so what are the triplets or triples 9 40 and 41 now do the next one 4 + 3 x squared 4 squared is 16 3 squared is 9 difference between these two you get 7 double of these - 4 times 3 12 12 times 2 24 at these two you get 25 so what are the triples they're 7 24 25 this one 6 & 5 6 squared 36 5 squared 25 now take the difference when you subtract you get 11 then double of these two 6 times 5 30 30 times to 60 then add these two you are getting 61 so what are the triples 11 60 61 next one 7 + 5 7 squared is 49 5 squared is 25 take the difference difference is 24 double of these - 7 times 5 35 5 times 270 so when you add these two you are getting 74 so what are the triples 24 70 74 miscellaneous exercise first one the chord a B of the circle with center or which lies at a distance of 9 centimeter from o is of length 24 centimeters find the radius of the circle so we will draw the diagram there's a circle it's Center or the chord a B so we draw called a be somewhere there which lies at a distance of 9 centimeter from oh that means 9 centimeter from oh the perpendicular length and the length of the chord is 24 that means 12 and 12 we know it's bisecting find the radius of the circle so radius if we take this - are you can use Pythagoras theorem because this is a right angle triangle so R squared is equal to 9 squared plus 12 squared 9 squared 81 12 squared 144 when you add you get 225 when you take the square root of 225 we get 15 centimeter father radius question number to construct the triangle ABC we are a B is 2 B C 3 and B is a right angle using the triangle you constructed find the value of root 30 okay so we'll draw the triangle B angle is 90 ABC a B is 2 B C is 3 find the value of root 13 so we'll see what is AC squared here this is construction so we need to construct the diagram but when you use the Pythagoras theorem just check what is AC squared 2 squared plus 3 squared 2 squared is 4 3 squared is 9 9 plus 4 13 so if you can find out the length of AC its root 30 now we'll construct the diagram and see what's the AC length so we need a ruler and we'll take the compass to construct the line so here so first draw a line and take the point as B BC length is 3 centimeters so we'll take 3 centimeters from here mark the point C take the compass measure three centimeter leg keep on top of this sand mark the point C that's C point then 2 centimeters we need take the compass again and Misha 2 centimeters and we need to draw a knock somewhere up now we need to draw a perpendicular line so we'll take the comm percent try to draw the perpendicular line to be let's take any length draw an arc take the other side and draw a knock then we'll take the point here take a longer distance and keep on top of this point draw an arc here and keep on top of this point and draw the other arc so the intersecting point and this one we can connect and here this point is 2 centimeter away this and this you get a C now we'll measure and see AC length with the compass and see what's the length so this is roughly smutch so you keep the compass on top of the ruler and measure the length you get three point six centimeters so correct to one decimal place this is three point six centimeters question number three construct straight-line segments of the lens given below so this is like Archimedes spiral so root eight we need to find out so rotate how can we get root eight so they here we don't need to measure the length so you don't need to draw it accurately so 8 root 8 how can we get 8 if we can get 2 & 2 what's that you get this side it says ABC triangle what is AC squared 2 squared plus 2 squared 2 squared is 4 4 plus 4 you get root you get 8 so if this is AC this is rotate so that's how you can get rotate so you need to take a right angle triangle with two and 210 how can we get 10 how can we get 10 3 & 1 3 squared 9 1 squared 1 3 plus 9 plus 1 10 so we can get if this is ABC triangle AC squared is 3 squared plus 1 squared 3 squared is 9 9 plus 1 you get 10 so AC is square root of 10 41 how can we get 41 so five squared that's 25 41 - 25 you get 16 so that means Oh so 5 squared so we will take ABC triangle so AC squared is equal to 4 squared plus 5 squared 4 squared is 16 plus 25 you get 41 so AC is the value 4 square root 41 question number 4 ABC is an equilateral triangle D is the midpoint of faming E is the midpoint of CD prove that 16 a a squared + 7 a b square we know when you draw the perpendicular here the midpoint and this one it's perpendicular so this is 90 degrees now we'll use the Pythagoras theorem for this triangle what is a e squared a d squared plus de square now we need a B a B is same as B cos C so what is ad ad is half a b de t is half of sating so when you simplify you get a B squared over 4 plus 1/4 of C d square so what is C D square using the other side so this is 90 degrees I can write C d squared is B C squared minus BD square but I know that BC squared B C is same as a B so I can write instead of BC square that's a B Square B d squared B D how can I write that's half of a B I can write down half of a B Square now we'll simplify I'm getting a B squared over 4 plus 1/4 of a B squared here minus half into 1/2 1/4 of a B Square now simplify inside first this is a b squared over 4 plus 1/4 inside this is for a b squared over 4 4 minus 1 3 a B squared over 4 now simplify I'm getting a B squared over 4 plus 3 over 16 a B Square now the common denominator is 16 lowest common multiple so that means I have to multiply this by 4 4 times this for a B squared plus 3 a B squared I get 7 a B squared over 16 cross multiply what's this this is AE square when you cross multiply I get the answer 16 Y squared is equal to 7 a B squared question number 5 in the triangle ABC B is an acute angle the foot of the perpendicular drop from A to B C is X prove that AC squared is equal to a B squared plus b c squared minus 2bc bx so this is done only for acute angle you can do this for obtuse angle as well then this becomes plus so we call it a Polonius theorem so you will get this in combined maths a-level class so we'll look at this one a C squared how can I write AC squared I can write x squared plus XC square I want to collect BC instead of XC how can I write XC x is VC minus BX open brackets ax squared plus BC square middle term is minus 2bc VX plus BX square know what is ax squared I'm taking this one here ax squared plus BX square and then b c squared minus 2bc B X now what can I write for this one ax squared plus BX squared that's a B Square so I'm getting a B squared plus b c squared minus 2bc B X so this is done only for acute angle if this is obtuse angle so you get something like this so instead of - you will get plus two times b c bx if this point is X so you can try that also we did this for only acute angles in this and we covered quite how to use Pythagoras theorem in real-life application and Pythagorean triples so remember how to get Pythagorean triplets or triples and apply Pythagoras theorem in real-life applications