Oh [Music] [Music] okay so start this off we're having a look at working out the length of a EB I've got two different triangles here two different questions so let's identify where a B is on the first one so a 2b is this length here now when it comes to right-angled triangles we've got no angles involved we can use Pythagoras to work out the missing lengths so Pythagoras theorem is a squared plus B squared equals C squared now that C represents this longest length here which we call the hypotenuse so this is the hypotenuse hyp so the hypotenuse and we've got the shorter sides a and B so four and seven so to work out this length of this hypotenuse we're going to do four squared at seven squared and work that out on the calculator so four squared plus 7 squared is 65 now that is obviously the length of C squared so to find out what C is we just need to square root our answer so to finish that off square root of 65 which gives us a decimal for this particular answer and that gives us a length of eight point zero six two two five and some more decimals and again if it asked us around it's a one or two decimal places we'd do that in the question now move it on to the next one we've actually got the length of the hypotenuse this time so we've got 13 and we're trying to find out them this length this time so A to B this time is one of the shorter sides so rather than add them together like we did down here we're just going to subtract them so we're gonna do 13 squared takeaway this shorter side which is 12 squared and again we're gonna square use our answer this is quite a good ones is actually one that could be done without a calculator we get 169 for 13 squared take 144 and 169 takeaway 144 is 25 okay so as you can see that's giving us a square number when we finish this question off to do the square root of 25 we get the answer 5 this is all in centimeters that'd be 5 centimeters so over here our answer would be 5 centimeters okay so Pythagoras square the sides add them together before square rooting for the longest side and subtract them for one of the shorter sides before square rooting okay so working out the size of ungulates again I've got two here and we're looking at parallel lines as you can see by these arrows here these lines are parallel so there are three rules that we're gonna really look at when we're looking at parallel lines as well as all on another or normal angle rules we've got alternate angles corresponding angles and Co interior angles so if I look at this one on the left here 110 there's lots of angles we can find now so all around this point here we've confined all of these angles vertically opposite 110 is also 110 and then next to the hundred and ten on the straight line there remembering angles in a straight line up 280 these would all be seventy there we go so it's up to you really how you actually find this final one I'm gonna have a look at finding this as edge shape here which is an alternate angle and alternate angles are equal so this 70 here is the same as this 70 here so X would equal 70 degrees in this one and we've no one been asked to give our reasons there so my reason may have been angles on a straight line equal 180 and that's all got the seventy and then I would have said alternate angles are equal and that's really important the way you say that so I'll turn it angles are equal okay there are also other ways that I could have done that if I split to another color we could have said okay well if this is 110 well then this here is also 110 because those angles are corresponding and again I'll try and draw this on a different color but they make this F shape okay so that hundred and ten and that 110 are corresponding angles so we could have done it that way and then done angles on a straight line let's look at this other one so 105 degrees down in the bottom here so straight away just above that we can find that this angle 75 because I was on a straight line link 190 now if you have a look we've got a triangle in this question now there's these two symbols here which tell us that this is an isosceles triangle and base angles and isosceles are equal it's underneath these two little Y cool little eyebrows okay so underneath this will also be 75 because base angles in an isosceles are equal we can then find this missing angle here because angles in a triangle like 180 so those two 75 seek 150 and if we take that away from 180 it leaves us with the other end that triangle being 30 there we go and there's all the angles that we found just using basic angle facts now to get that angle X is a couple of ways we could do it we could find this angle here and then in two angles on a straight line or we could again identify that actually we've got one of these alternate angles going up making this set shape again so that's an alternate angle with the 30 so that's 30 degrees as well so for this one we'd say x equals 30 and again we'd probably give this same reason over here that alternate angles are equal unless of course we took a different approach which we could have done again a lot of different ways you can tackle these questions but a few there that you need to make sure that you're trying to remember so working out the size of the angle between this regular octagon and regular hexagon so there's different ways that you can approach this but we need to know what the angles inside a hexagon all add up to and what the angles inside an octagon add up to different ways of doing this you can do the 180 let's draw so you can keep having 180 so three sides is 180 four sides is 365 sides is 540 they can keep doing it this way or you can apply a bit of a formula so I'm gonna go to go for the form will even off started writing this down but that is an option there that you can use so for six sides and a hexagon we take away 2 so 6 take away 2 which is 4 and then 4 times 180 tells us the total amount of there should be a time sign let's change that four times 180 gives us the total amount of angles in a hexagon which is 720 so all the angles inside a hexagon equal 720 and there are six equal angles in there so if we divide that by six 720 divided by 6 gives us 120 degrees so all of these angles in the hexagons 120 obviously we're concerned about this little area here in this question we're gonna find the angle between them but we've got the Anglin hexagon that's really important when you see these words regular hexagon and regular octagon always just work out the interior angle and label it all over the diagram let's move it on to the Octagon same process eight sides take away two six and then times up by 180 they cone six times hundred eighty which is okay if you have a calculator if not you have to wear that out but six times hundred eighty is 1080 and again it's got eight equal angles so if we divide that by eight one thousand and eighty divide it by eight gives us 135 degrees and again we can just label that all over the diagram so hundred thirty-five hundred eighty-five and lots of these but I'm concerned again with this little area here so angles around a point equal 360 and at the moment we've got two of these angles kinda like a jigsaw so if we add these two together 120 plus 135 at the moment we've got 255 degrees and we need it to add up to 360 so to finish this off if we do 360 and take away that 255 that we've just worked out there we can missing angle of 105 degrees and that's our final answer there 105 degrees which we could always label on the diagram as well 105 and there we go so where have you seen these questions where it says regular octagon regular hexagon okay to spot these what like these bits of language here just work out their interior angles and label it all over the diagram so I'm looking at circles these two formulas that we need to know circumference equals pi times diameter an area equals PI R squared remember the units for area have squared in don't they when we use them to meet the script centimeter squared so I just like to remember that little formula for area there is the PI R squared the one with the squared in so we're gonna work out the area and the circumference of this circle so to work out the area we're gonna do PI R squared if we start with out so the area equals PI R squared and we just need to know what the radius is over here the diameters a so we're going to first half that for the radius so I'll just write R equals four and we'll just gotta type that in so are we 10 pi times 4 squared 4 squared is 16 so it'd be Piton 16 and you can just write like a 16 pi so if it asks you to keep your answer in terms of pi you can just do that the water-type when i calculator if not so 16 times pi will give us a decimal which is 50 point I'm gonna round this but obviously the only round it if it says to in the question be 15 point 50 point two seven if around centimetres squared and that'd be an area there to two decimal places let's have a look at working out the circumference so circumference up here is pi times diameter now I've been given the diameter is 8 so to get the circumference we just do pi times 8 and we could just say 8 Polly so we could just say 8 PI which just means pi times 8 but again if it asks us to work it out we're going to be pi times 8 on the calculator so 8 pi press equals and we get and if I round this to one decimal place this time I get 25 point 1 and circumference is a distance it's around the outside we've just centimeters 25 point one centimeters but again rounding it to how the question asks and remembering you can write it in terms of highlight down here 16 pi and I'm here 8 pi as well okay so working out the area of a circle sector now we're gonna do this in a very similar way to what we did with the circle we're still going to do area equals PI R squared but because it's a fraction of a normal circle we're going to times it by whatever fraction of the circle that is so it says over here is 102 degrees so it's a hundred and two degrees out of a normal 360 degree circle so I'm gonna plug the numbers into there and we'll get our area so system here the radius is 8 which is quite nice so it's pi times 8 squared multiplied by this fraction of the circle 102 over 360 you can type it into the calculator so pi times 8 squared times running two over 360 and that gives us an area here of 56 point nine six seven five four six seven nine and I tell you what let's round it to two decimal places so that would be 56 point nine seven and again units centimeter squared and there's the area of the sector again you could actually work out this here as well if you're asked to it's called the arc length so it would do the same thing but we'd use our circumference formula so we'd do the circumference equals pi times diameter multiplied by the fraction there which would be a hundred and two over 360 so we could work out both just using the appropriate formula in the same way all right working out the area of a trapezium there's a formula for the average easy emits a plus B over two times by the heights or it can be written in a slightly different way you can say half a plus B again times in by the height now a and B are the parallel sides which in this case is the five and the knowing so we add those together this start with so five plus nine we have that so five plus nine is 14 half of it is 7 and then we times that by the height and they've just got to be careful here because you are given these two lengths on the side and they're really there just to throw you there not the height their diagonal lengths this four in the middle is what we want the perpendicular height from the base indicated by that red arrow there so if four is our height so we never do 7 times four which gives us 28 and it is an area since centimeters again so centimeter squared says be careful the trapezium there that you do identify the perpendicular height from the base so but work out the surface area of this cuboid now well there's only three surfaces we can see we've got this one on the front but there's also one on the back that's exactly the same back here that we can't see we've got the one on the top number 2 which is the same as the one on the bottom and we've got this one on the side here which is the same as this one that we can't see on the side so if we work out the area of these three faces we can just double our answer to get the ones on the back as well so I'm gonna do this in steps I'm going to do the face number one which over here is eight times 12 so eight times 12 is 96 centimeters squared face number two the one on the top we've got 14 along there and it's not labeled but this is 12 as well the same as the bit below just down there so that's 14 times 12 so 14 times 12 you just gotta work that out and that's 168 this was 168 centimeters squared and then the last one face number three on the side it's not labeled but we can label at the heights a and the lengths going backwards is 414 the same as on the length on the top there so we've got eight times 14 so eight times 14 for our last one thick times 14 is 112 112 centimeter squared and if we add all of these up 112 plus 168 plus 96 we get the answer three hundred and seventy-six centimeter squared they just got to be careful at this point obviously don't forget there are two of each face so we just need to double our answer all times that by two so times up by two gives us a total surface area here of 752 centimeter squared and there's our final answer for the surface area working out the volume of a cuboid is quite nice and simple I do like to do in a particular way though just so it's the same for all these shapes where we have a constant cross section so this cross section I'm gonna have a look at this front face just here some photos gonna just work out the area of that which is 8 times 12 so 8 times 12 gives me 96 that is the area of the cross section gneissic centimeter squared I don't to get the volume which is times it by however far back the shape it goes which is the same as that 14 right there so times up by 14 96 times 14 will give us our volume - 96 times 14 is 1344 it's a volume so we put centimeters cubed and that's how you work out the volume there well it's for any sure we've got a constant cross section if you imagine sometimes we have these triangular prisms and sometimes we could even have a trapezium face prism there okay anything with that constant cross section you always work out the area of this front face to start with the cross section and times it by the distance it goes back that same logic can be applied to a cylinder so our cross section this times a circle so we've got to do is work out the area of the circle remembering area equals PI R squared so working out the area of that circle it's pi times 4 squared which again 16 pi we can write that as a decimal and what I'm gonna do is I'm just gonna leave it as 16 pi for the moment so I don't have any rounding errors here but you can write it as a decimal at this point but I'm gonna leave it a 16 pi and we've got the area of the cross section you just times it by how far it goes through the shape which in this case is this 15 so what I've got to do is times up by 15 so 16 pi times 15 which comes out as 240 pi which is also a decimal so it just depends on how the question once the answer here so it's 240 PI or if I write that as a decimal remember that Esther D but no F to Debus on your calculator depending on which one you have yet 753 I'm gonna round it to two decimal places again 0.98 centimeters cubed for volume and there's the volume of my cylinder so surface area of a cylinder is a little bit more complicated because we've got these two circles to work out and then we've got this shape wrapped around the outside of the cylinder so the area of the circle would actually already works out pi times R squared is 16 PI and that equals the circle okay and not forgetting we've got two of those so it's times up by 2 which gives us 32 pi a chicken is a number but I'm just gonna leave it till the end for the area of my two circles now the next thing that I have is this sheet wrap around the outside and if we were to unravel that we'd have a nice big rectangle and the height of that rectangle there is 15 the length of the rectangle is actually the circumference of the circle so we've looked at circumference of a circle the circumference is pi times diameter so I'll do 8 times pi or 8 pyv the diameter there being 8 it's giving us a radio of radius of 4 so we could put the diameter equals 8 that's 8 pi for the length and we work out the other rectangle just by doing length times width so to work out the area of this order 15 times the 8 pi and that gives us 120 pi for the area of that rectangular part wrapped around the outside and we've got to do to finish this off he's add these two together so we have 32 pi add 120 pi and in total that gives us 152 pi again we could just write this as a decimal 150 2 pi comes out as 477 0.5 to one decimal place centimeters squared for surface area and there's our final answer so you just gotta remember you obviously got the circles top and bottom and then to work out the area of the curved part you need to think about that circumference there being the length of the rectangle okay so we've got these two shapes are mathematically similar work up length of a and B now that they're mathematically similar means one is an enlargement of the other which means there's a scale factor between them so if we can identify a similar side on both of them which we've got here four and ten we could find the scale factor so I always just do the big one divided by the little one so 10 divided by 4 and 10 divided by 4 4 fits in twice and 1/2 4 8 and then an extra 2 so it's 2.5 you might just get a whole number there but there's my scale factor and I always label that I just say SF so the scale factors 2.5 so to get from a small and let's have a look at a to start with we've got to get from 7.5 back down to a so that's going from the big shape down to the little one so to get from the big one down to the little one we divide by the scale factor so divide by two point five and seven point five divided by two point five will be three centimeters and that's actually going to show the same for this other one here look because this one's three as well but to get from three to be we actually just times by that scale factor so times by two point five and three times two point five which we already have there is seven point five centimeters okay so you just got to remember divided from the big to the small one and then multiplied by that scale factor to get from the small to the big one it's all about identifying that scale factor that we did appear that two point five there we go working out some bearings so write down the bearing of beef from a so if I'm a B and I want to go from A then I'm standing over here at Point a now there's three rules for bearings they're always measured in a clockwise direction there's always three digits I always measure it from north okay so they are my three bearing rules so looking here if I'm if I'm standing facing north which is shown by the North line I turned clockwise I'd have to turn this rule 60 degrees so that's thing there 60 degrees but it doesn't have three digits so I stick a zero in front of the 16 so B o6o degrees or naught 60 degrees now the next part the question says write down the bearing of to a from B so that means now I'm standing at B so let's just get rid of these little bits so this time a from B means I'm standing at B obviously following these rules if I face north I've got a turn all the way around this way to get back to the direction of going back to a so let's have a look at how we can do this now what I quite like to do with these sorts of questions is to think about if I can extend the line slightly so if I extend the line because these two North lines are parallel we've actually got a corresponding angle there so this bit here is actually 60 degrees so we can think about other angle rules while we're doing this so from that bit just down to my little dotted line I've put in a 60 degrees because this is a straight line this extra bit here is an extra hundred and eighty degrees so actually what I've got there is 60 to get me down to here and then an extra hundred and eighty if we have those together 60 plus 180 we get a total of 240 degrees some answer for that would be 240 degrees which does have three digits so kind of leave it like that of course there isn't of the way that you can look at this as well we could think about these two angles here as being Co interior that add up to 180 these two out of 280 so this one here must have been 120 so I actually I could take away 120 from 360 because that full turn there is 360 and we've got this little bit here as being a hundred and twenty so could actually do it like that as well so there's two ways that we could approach it okay so translating a shape by a vector now that top number in a vector means left and right and the bottom number means up and down okay think about positive numbers moving towards these positive numbers on the axes and negative numbers moving towards the negative numbers as well so four moves it right towards the negative numbers by four so all I ever do is pick a point you can pick any and I move it right by four so one two three four it's going to end up there the next thing I do is look at the other number which is minus two now negative two moves down towards the negative numbers so one two down and that means that new point is going to end up just a so all I have to do is draw this triangle in exactly as it is from that top point so it goes down three and across - and there we go we can just draw it in like that not forgetting as well you might actually be asked to describe a translation in which case you'd say it's like it says above you say translate the shape by a vector I need to say all the vectors there is okay so reflect the shape in the line x equals 1 now this is your x axis and this is your y axis so x equals 1 is right there and the only line that you can do from that point other than going across the x axes and doesn't see the XA the x axis is to go up and down so x equals 1 is this line here if you think about any coordinates on that line the x coordinate is always 1 so that's our reflection line there and we're going to reflect it in that line which is quite nice and easy to do pick a point and we could go 1 2 to the line so 1 2 away and just follow that process for all the points so picking this point that is far away so another fall away gets me to there I'm the same one at the top that's far away and 4 away against meter there and then just joining it all in obviously using a pencil for this one and that would be my triangle drawn in particularly neatly there we go okay not forgetting as well then you could have a line it could say the line I don't know y equals 3 which would be across at y equals 3 which would be across here there's another couple of lines that we could have as well we could have this one y equals x the line y equals x is the diagonal line here where all the x and y coordinates are equal so we could have that one as well where we have a diagonal there's one more that we could have as well let's pick a different color for this we could have y equals a negative X which is very similar to the one above but it's pointing in the opposite direction and if I do that in blue as well that's going down this way so we could also have to reflect in a diagonal line they can't you just follow the same process counting direct billing okay so rotate the shape 90 degrees clockwise about the point minus one zero so first thing to do is obviously to locate that point which is minus 1 across nuts up and down so minus 1 0 just there the next bit this is a lot easier if we have some tracing paper so you stick your tracing paper nice and flat over your shape trace it in so trace over your shape stick your pen on that rotation point and just rotate the tracing paper not moving it away from the pen so we would rotate it 90 degrees so it would be facing this way and just making sure we could stays nice and flat and it would rotate the shape let's have a look it would go to here on your tracing paper and then you would just lift your tracing paper up nice and carefully and draw in at that spot and again the same process if you have to describe one of these you stick your tracing paper over the top and just move your pen into different points pick in places like here and here and moving it around until you get the one that matches when you switch your suppose all your paper around 90 degrees and remember and obviously in your description you need to put the amount of degrees the direction that it's gone in and that rotation points along with the words rotation okay so enlarge the shape by a scale factor of two from the point minus four minus three so let's identify that point first minus four minus three and then we're just going to vote enlarged by scale factor of two so what I do is I pick a point on the shape normally the closest one say this one here and I just think all right how do I get there from the point so we go to a cross I'm one up and I'm just going to do exactly the same again so two across one up and that there is a scale factor of two the first movement to the shape is my first scale factor the second is my second scale factor there so if I had a scale factor of three I'd do the same again I'd go to across one up and that be a scale factor of three but this one has just set a scale factor of two so I'm going to leave it there now if I have a look at the shape that's drawn originally it's too long it's three up and this is gonna enlarge it bass cavity which doubles those numbers so it's no longer gonna be three it's gonna be six and it's no longer gonna be - it's gonna be four so from this point here that we've got - all I need to do is redraw that shape in remembering it was the bottom left corner this one here so I just need to go forward across one two three four and six up one two three four five six and then join it all up nice and neat with the pencil and ruler and there it is enlarged by a scale factor of two just remember if you have to describe one of these it's a really nice way of doing it as well you just pick the two corners get your ruler and a pencil join them up really nice and neat pick another one match it with the appropriate one and then again join that really nice and neatly and it show you here down here look where the enlargement point came from and you can find the scale factor by she's looking at the sides so the side length is 6 here and the side length is 3 there and 6 divided by 3 gives us a scale factor of 2 so if it was already drawn in we found our enlargement point would you say it's an enlargement a scale factor to and from this point down here just like it did in the question enlargement scale factor 2 and from this particular point geometry so we're gonna start off with this volume of a frustum now you would be given the volume of a cone formula and that is from volume equals 1/3 PI R squared H where R is the radius and H is the height so if we look at this frustum here the only thing it doesn't give us is this length across the top but if we have a look the height of the little cone on top is 20 and the height of the big cone if it was still there is 40 so was half the size of the diameter there is 15 now in order to work out the air volume of a frustum you gotta work out the air volume of the big cone if it was there and we're gonna take away the little cone that's been chopped off the top so I'm going to start off with the big one I'm gonna label this the big cone and I'm just gonna plug the numbers into the into the into my calculator so it's one-third times by PI times by the radius squared now careful because it's not 30 it's 15 half of that so 15 squared times by the height which is 40 so typing that all in to the calculator and I can leave that in terms of pi or I can write out the full decimal and it comes out as nine four two four point seven seven seven nine six one centimeters cubed now for the small cone on top there we go we can work out the volume of that as well so it's one-third times pi times the radius squared careful's it's half of 15 so 7.5 squared and times that by the height of that cone which is 20 and that gives us a volume and again if we just type that in let's have a look so one-third x pi x 7.5 squared times 20 and that gives us a volume again I could leave it in terms of pi but one one seven eight point zero nine seven two four five and again that centimeter cubed so in order to work out the volume of what's left when the little cones been cut off the top we just need to do the big cone subtract the volume of the little cone there so I'm just going to subtract this away from that so subtract and again I'm just gonna do that on the calculator so nine four two four point seven seven seven nine six one take away that answer there leaves us with the total volume of eight thousand two hundred and forty six point and I'm going to round it to two decimal places or that you would be asked out rounding a question I'm gonna go for a point six eight centimeter cubed just be careful what the question asks office if you said one decimal place it'd be point seven it might even just say fourth or three significant figures in which case you just gotta make sure you just round out how the questions asked but there you go that's using the volume of a cone formula on on a frustum now it's a little bit more interested in just doing the normal comb makes one here we have a hemisphere and it says work out the volume of the hemisphere now again we'd be given the formula to work out the volume of a hemisphere and volume of a hemisphere is 4/3 PI R cubed and that is our final volume they're all mismo slices that's a rubbish 3 let's change that one 4/3 PI R cubed so we got to do is stick the numbers in the formula again so if I want to work out the volume all I have to do is in the calculator I do 4/3 x pi x 8 cubed and we'll get our answer here straight away so fraction button 4/3 x pi x 8 ^ 3 and we get a volume here of 2 1 4 4 0.66 again if I round it to two decimal places centimeters cube just remember obviously to write all the numbers down and only round it once you've been asked to okay could be has to work out the surface area though so I'm not included it in here but again you'd be given the formula for this it's all to work out the surface area the surface area of a sphere okay so once you've got your total volume of your sphere there just remember this is a hemisphere so it's half the size of a sphere exactly half so what I just need to divide my answer by two here and once we divide that by two we get a final answer of let's have a look one zero seven two so 1072 0.33 again I've rounded it they're centimeters cubed and that would be your final answer here for the volume of a hemisphere not forgetting as well you could be asked to work out the surface area and if you have to work out the surface area again it gives you the formula of a sphere and the air surf service sphere is 4 PI R squared so again if I was to work out the surface area of this one it's a little bit more complicated but you've only in the sense that we work out the full surface area so we do 4 times pi times 8 squared I finish to go for this that's work it out 4 times pi times 8 squared that gives us the surface area of a sphere being 804 and it's 0.25 as well again we'd have to have it because it's half okay so half of that would give us a surface area of 402 0.12 and that's obviously a surface area centimeter squared but if you're working out surface area you've got to make sure you don't forget as well the circle sitting on top back is the total surface and the area of a circle is PI R squared so we can work out the area of the circle as well if we just do that pi times H squared we get an area of the circle there of 200 and 1.06 and you would just add these two numbers together to get the total surface area so if you add 200 and 1.06 we get a total surface area o 102.1 202 that gives us 603 0.18 centimeter squared again so I know we're looking at volume there but just thinking about if you have to do the surface area as well I need approach that when you've got a hemisphere and you've got to make sure you have them and add on the extra circle okay so we got some similar shapes to cones I mathematically similar the height of cone a is 4 and high okay is ten they can always imagine what this is going to look like you can always draw yourself a picture if you haven't been given one and we've got a bigger coin so we've got a height of the smaller one being four and the height of the larger one being ten now straight away you can work out a scale factor between these two if they are mathematically similar which means one's an enlargement of the other so I can do the bigger length 10 divided by 4 and it gives us a scale factor of 2.5 now when it comes to scale factors that is our length scale factor and always label it length scale factor now there's two more scale factors we could out we could want to look at and that's the area scale factor or the volume scale factor and to get from the length scale factor to the area scale factor you square it okay so we square the scale factor to get from the length scale factor down to the volume scale factor you have to cube it so if we just have a look at this question here it says the volume of cone a is 40 work out the volume of cone B so that's cone a is the smaller one in this case I should probably label that a and B and this is the volume of this one is 40 so in order to find my volume scale factor I need to do 2.5 and cube it to get a volume scale factor so go do that on a calculator 2.5 cubed gives me 15 0.625 and that's my scale factor in terms of the volume so the volume of cone B is going to be fifteen point six two five times bigger so in order to get that volume there but it's gonna multiply 40 by fifteen point six two five so it's times up by 40 and we get a volume here of 625 centimeters cubed they go just remembering as well if we were going from the bigger down to the smaller we did divided by the scale factor but as we getting smaller to bigger we're gonna multiply so I'll get a different one so similar question two cylinders are mathematically similar we've got cylinder a and cylinder B nice a cylinder a 116 cylinder B is 20 so Haysom a larger one suppose draw a little basic diagram of this but got a and we've got B there we go and this time it says it actually gives me the volumes so it says the volume of a is 160 now and the volume of B is 20 so again if I find a scale factor here I can do the bigger one divided by the smaller one 160 divided by 20 gives me a scale factor of 8 now thinking about this area scale factor length scale factor and volume scale factor there we go I've got this time I've got the volume scale factor and that is 8 but I can't get from volume straight to area now if you remember to get from lengths to volume we had to cube it okay so to get back from volume to length we have to cube root it so I did the cube root of 8 that tells me the length scale factor is 2 not the question here is saying in then starts it asked me to work out the surface area so I need the area scale factor and if you remember just from the last one to get from areas lengths onto area we square it so that'd be 2 squared which before some an area scale factor is 4 okay so whenever you're given an area or volume you have to give back to length first and then get back to in every or or volume so my scale factors fall for the area and it says the surface area of cylinder a is 40 so that's the bigger one I'll just write area of the surface area so to get from the bigger one down to the smaller one we're gonna have to divide by that scale factor which is 4 for our area and 40 divided by 4 gives us 10 so we get a surface area of 10 centimeters squared all right there we go there's think about some similar shapes okay so we've got a transformation here it says in large shape P with a scale factor of negative 1/2 with the centre of enlargement 0 0 so it's mark out your centre of enlargement first and an enlargement with a negative scale factor and a fraction here means that we just got to do this very very carefully now the first things first you need to pick a point on the shape and just figure out how you get from the centre of enlargement to that point there and if I just count that that's one two to the right and one two three up now all I need to do to do this and it's got to be quite small in this diagram is I just count in the opposite direction by whatever the scale factor is that's what the negative is so rather than going from from zero zero rather the game rights enough I'm going to go lift and down but it's a scale factor of 1/2 so needs also have those distances just as if it was negative - I would double those distances but for negative 1/2 so I'm gonna go left one rather than 2 and I'm gonna go down rather than go up 3 I need to go down 1.5 or one and a half so one and a half gets me to there there we go so that's one and a half I'm gonna do the next one in different color so I'm gonna pick this point here and let's just think how we get to that let's get rid of some of these markings here there we go so to get to that we go 1 2 3 4 across and 1 2 3 up so I'm just gonna have that again I'm gonna go to left rather than fall right so 1 2 and then that one and a half down which again gets me down to there there we go and that is at that point then again I just need to repeat the process for the last one you might be able to start to do these in your head when you've got some practice but if we do the last point here being the top one I just see how I get there so it's one two across just like it was - that red one and then we got to go up 1 2 3 4 5 6 7 so half our that's going to be 3 point 5 so I'm going to go one to the left ok so we need to go down three point five one two three and a half just there there you go obviously you can do all this in pencil to keep this all nice and tidy but when you've got a negative scale factor you're just gonna go backwards in the opposite direction but whether that scale factor is obviously join it all up nice and neat and there we go and that is that one there just always go back and double-check so it was seven up so we went three point five down that's absolutely fine the other one was three up when we went one point five down hard for that and the other one there perfect okay so you can see with these it does actually get bigger and it goes bigger or smaller and it rotates 180 degrees and that's how a lot of negative enlargement does to a shape okay so we're going to have a look at another one here okay so it says described the single transformation that maps shape be on to shape a this is quite nice hopefully you can tell that it's an enlargement okay cuz obviously one's got bigger than somewhat bigger and smaller and we're going from B - I know what you can do is you can get your ruler and I've got to think of everyone's rafter to run do this quite carefully but you get your ruler and you join up the similar points now the similar points are here and here okay obviously has been rotated a hundred eighty degrees and I forget you're all are enjoying that up you end up with a little line like that and then you just do that for all the other points so I can probably get away with just doing two we'll have to wait and see so for joining up these two similar points here there we go they join up just like that and what you'll find is it this little crossover point and that tells you where the center enlargement is so I know that it's an enlargement we've already got that so I'm gonna have to state it's an enlargement okay so it's an enlargement I've got to get the scale factor is I haven't got that quite yet but I'll have a look we know it's negative as it's getting this one eighty flip so you know it's negative will come up with a number in a sec and I know where the center enlargement is so centre of enlargement let me just write down what that chord and it isn't it it's two two so since our enlargement to 202 count that scale facts are just going to count this side here maybe on this side here and just see how much bigger it's got so that's come from a length of 1 to xi of 3 so that is a negative 3 scale factor so it's an enlargement scale factor negative 3 centre of enlargement to 2 I can use that approach for any of these enlargements whether it's negative or not also you might just want to note here that if it had have been from A to B it would have been the other way around it'll come from 3 down to 1 I'm a scale factor instead if I was going for me to be it'd be exactly the same description there but my scale factor would be negative 1/3 because it's getting a third of the size going the other way okay so two different ones but the description would be exactly the same there okay so can I look at some circle theorems the purpose keeping this video as short as we can I'm only gonna go over a couple of them and it's how to approach the question if you want to have a look at every single circle theorem how to tackle it without type a question you have to look at my circles theorems video now some information to go along with this I'm gonna say that AP and BP are tangents okay so when it comes to tangents they meet at equal length that's one of our circle theorems so these two lines here our equal length meaning the APB that triangle there is an isosceles meaning that the base angles these two are the same so there we go we can start to figure out some of the angles here because if we do 180 the angles in a triangle take away 86 it leaves us with 94 that we can split in half to share it between our two triangles there so 94 divided by 2 is 47 so both the angles at the bottom of the triangle there are 47 there we go and as with all these questions it always says to state your reasons so you would say tangents meet at equal length therefore this triangle is an isosceles okay and writing that down okay on to the next bit we've also got and it doesn't doesn't say here but oh is the center of the circle now Oh to be there is a radius looking at this line just here and a radius meets a tangent at 90 degrees meaning can always draw this on the diagram when you see a tangent meet and a radius we've got a right angle just here so we can work out the value of x because if the full angles 90 we're going to 90 take away the 47 and that leaves us with 43 degrees the important part with this question is to make sure you write down all those reasons so we would write the tangents meet at equal lengths isosceles base angles are the same and then our final reason for this bit of the working out the tangent meets the radius at 90 degrees and therefore we could do 90 take away 47 okay a different circle theorem question with some different circle theorems within this one so if you have a look we've got these points a Dec be around the circle and that forms our quadrilateral and these are called two cyclic quadrilateral 's and the rule here is that opposite angles add up 280 so we've got the seventy over there so opposite that is the nine is the Y and opposite angles have to add up to 180 so we could do 180 takeaway 70 and it gives us our answer of 110 degrees and our reason for that which we would have to write down again is that opposite angles in a cyclic quadrilateral add up to 180 the next one here is to find this angle X and that's going to involve one of our other circle theorems and when you've got these points made at the center I'm just old like a point D and B I always do this with a little highlighter but if I make this angle 70 here I can also from the same two points make this angle here a center and that's one of our other circle theorems angles at the center are double angles made at the circumference from the same two points so to work out angle X and this first one we did was Y to work out angle X we would just do seventy times to double it and that would give us 140 degrees and again that would be accompanied by the reason angles at the center are double angles at the circumference when they're made from the same arc so our two answers here are 110 degrees and 140 again just a few little bits of circle theorems there to have people having a thinking about and making sure that you are writing down all those reasons that's absolutely crucial for these okay so we're on to some congruent triangles it says prove angle abd is congruent to CBD now if you have a look abd is the triangle on the left and CBD is the triangle on the right now in order to proof triangles are congruent we need to look for similar sides and similar angles now the first one that pops into my head is the line B to D this line in the middle okay I like that let's line B today in the middle is the same for both triangles so I could prove aside straight away so I can say a side I've got BD it's the same in both so both triangles already share a similar side or exact same side BD is the same in both remember we're trying to prove that they're congruent which means that they're exactly the same so let's get rid of that we could have a look at another one now can we see any other sides that are the same well I've got these little symbols and I've already got an arrow pointing to two of them really we've got these little symbols and all of them which means that all of those lengths are the same length so we could state any of these are the same now we've got a B this one here number one is the same as BC number two so I can do another side so I can say that a B equals BC and actually that's given to us in the question so I would just have to say given in question okay and actually we using that same logic there we could actually prove that these two down the bottom which are labeled three and four are the same as well because they're all the same length so I could prove another side so I could say that ad equals CD and again that's given in the question there we go that's what I have to do is state why they're congruent there and I can just say and I was right it's down at bottom I can say therefore they're congruent okay because the F because the it and the SSS are all here so side-side-side congruent and we're using the SSS rule okay there's other things that you can look for as well you Cavill it for angles obviously there's a few of these that you can use and it's just an idea of how to approach and how to lay it out just label them what you found but because these are obviously isosceles triangles so this one on the left here we've got two sides at the same length so we know the base angles are the same here therefore as they're all the same length those base angles must also be the same as the base angles on the other triangle so we could use that as well and if those angles are the same then this angle at a has to be the same as the angle at C so actually we could prove all of them so there's a lot of different types of questions here a lot different proofs but there's just an idea of one of them and how to lay it all out okay so gonna have to work out some missing links in this triangle now it's not a right-angled triangles if I'm missing lengths and angles we can use either the sine role of the cosine role and these are rules are you're gonna have to remember well I'm gonna have a look at one to start with and how we know when to use it now the first thing you do is you try and identify in a triangle for first of all what we're looking for which is a B so I'm gonna label that X now what I look for straight ways do I have pairs of opposites so I have this pair of opposites here and I've got both of those and then I've also got X opposite to this angle so I don't have X but it's in one of my pairs of opposites so when we've got this scenario where there's two pairs of opposites we can use the sine rule and we only ever need part of the sine rule so I'm just gonna use a over sine a equals B over sine B also equals C over sine C but we only ever use only ever need to use two of them here so let's have a look we need to label this up and I'm going to completely ignore the letters that on the actual triangle itself I'm gonna label this angle a which it already is and they say on this side little a opposite that and then this angle B and then one opposite little B and I'm going to do now is stick all the numbers into the formula so let's have a look a is 12 so it's a 12 over sine the one opposite that 55 is equal to B which is our x over sine B obviously you should already know there's two for two variations of this formula we could have it flipped over so we could have sine a very equal sine B over B but this is our one for side lengths and we know that because our unknown pieces on the top so we're able to isolate this now quite easily so sorry I've written B there it should be 20 there we go sorry there we are 20 so what we need to do is times both sides now by design 20 what you could do is work this out when your calculator and times your answer by sine 20 but I'm gonna multiply it straight over so I can type it all any one go so times by sine 20 and if we do that we get 12 sine 20 it goes onto the top there over sine 55 there we go and all we have to do is type light into a calculator obviously just being careful that you put these angles in brackets some calculators are gonna need you to do that so if we settled into the calculator again not forgetting you could just work out 12 over sine 55 first and then times it by sine 20 but I'm just gonna go for it like this so 12 sine 20 on the top closing your bracket over sine 55 and on the calculator just writing down what you got you got five point zero one zero three five three nine nine eight now a question would normally say how to round this so if we imagine it was two decimal places for this one it would be five point zero one and it's a length so centimeters okay so just obviously would just be careful the question says let's have a look at an idea whether we've got to find the angle okay so in this question let's have a look work out the size of angle BAC so let's identify that that is here BAC okay so we're gonna use the formula the other way around this time so sine a over a equals sine B does not say saying be signed B over B okay so plugging in our numbers let's just label it up so let's call this one a as the eggs next to it that's fine and again I'm just going to dry over this one I'm just gonna put B and B okay just cause the way I've written my formula so then sticking in all the numbers what have we got sine a is sine X so we have sine x over 20 equaling sine 43 over 14 okay so exactly the same approaches we did before we can isolate the sine X play x in both sides by 20 and again you could work that right hand side out and times by 20 but I'm just gonna go stick it up the top there so we end up with sine x equals 20 sine 43 over 14 if we type that into the calculator now what do we get 20 assign 43 over 14 and we get an answer here let's write it over here so we get sine x equals zero point nine seven four two eight and a few more decimals obviously just like normal trigonometry when you're doing sohcahtoa to get the actual X here we have to the inverse of sine so we're leaving that number on your calculator you do ship sine which gets you sine minus one type in that answer or just press your answer button so shift sine answer press equals and I get an answer here of seventy six point nine seven seven nine and again a question what our system round it here just depends so let's just go to the nearest degree we'd go for 77 degrees obviously just making sure what the question says here but let's just round it to one the nearest degree there so 77 okay so that's how you use the sine rule right let's see how these questions different then so work out the length of a B so this one over here now straight away looking at this look we've got a pair of opposites there but I don't have any other pairs of opposite opposite so I've not got anything opposite my fifteen I've not got anything opposite the twelve so I can't actually use the sine rule here and that is your clue that is your hint here that the sign rules not going to work we're gonna have to use the cosine roll so another rule that you need to know so the cosine rule is he squared equals b squared plus c squared minus 2bc because a okay so a being the side we're looking for so we'll label this little a and this one big a ignoring the letters on the triangle and then labeling the other two sides and they can be B and C however you like and from there all you got to do is stick these numbers in it's actually quite simple to use when you know it so a squared equals B squared which is 15 squared plus C squared which is 12 squared minus and agrestic this bit in brackets 2 times 15 times 12 cause a which is down here which is 20 there we go alright so sticking that all in the calculator let's have a look what we get so 15 squared plus 12 squared minus 2 times 15 times 12 cause 20 press equals and I get a squared equals 30 point seven one zero six and a few more decimals obviously that is a squared we don't want any squared or know where a is so we just need to square root both sides now so square root leaving the uncertainty calculator square root answer and we get a equals five point five four one seven one nine and again obviously you'd be asked to round this in a particular way in the question let's go for two decimal places so a equals five point five four centimeters all right there we go and there's using it at the cosine rule okay so working out the size of angle BAC which again is this one at the top and again just having a look there are definitely no opposites cuz we've got no other angles but the angle that we're looking for is gonna be our a and it is opposite nine there we go so the others can be B and C again now obviously here we're looking for an angle so every sub T if you choose to learn the formula I tend to find that I just learned this formula a squared equals B squared plus C squared minus 2bc kaze and then quite happy just rearranging that to get kaze on its own so in order to do that I'm gonna get this whole minus 2bc kaze I'm gonna add it to the other side so we get a squared plus 2 BC cuz a equals B squared plus C squared now I can get rid of that a squared so we can minus a squared from both sides and minus a squared and you get 2 BC cause a equals B squared plus C squared minus a squared now and then you can finish off this rearrangement you can divide by 2 BC just to leave you is cause a so cause a equals B squared plus C squared minus a squared over 2 BC so two you can choose to just learn that formula if you want but that's the formula we're gonna use to find angle so plug in all these numbers then just into my formula there we'll get cos a equals B squared plus C squared so it's in squared plus five squared minus a squared so minus 9 squared all over 2 times B times C so 2 times 10 times 5 nice and easy talk tonight into the calculator so a fraction button 10 squared plus 5 squared minus 9 squared all over 2 times 10 times 5 and that equals sour like so cause a equals zero point four four and then same process again we need to do the inverse of course so it close - one of your answer and you get let's have a look chief cause answer and I get 63 0.896 okay degrees and again we could around that so we could just say 63 point and let's just go to one decimal place sixty three point nine degrees again just reading the question and that's how to use the cosine rule for finding angles moving on to the area of a triangle so your over triangle formula obviously not half base times height because we don't have the height here so for any triangle the area is hard of a sign see so half baby signs see another formula there that you need to know and let's just have a look at how we apply that so signs see this time the angle that we're going to use is gonna be our C so we'll call that C this would then be little C and the others will be a and B and they can be a and B in either order and then it's as simple as just sticking those numbers into a formula so it's 1/2 times 5 times 8 times sine 41 there you go type it into the calculator so 1/2 times 5 times 8 times sine 4 C 1 and we get the answer thirteen point one two if a rounded to two decimal places it's area so that's meters square just being careful of the unit so 13 point 12 meters squared there we go there's using the arrow triangle formula question like this though you might be given the area of the triangle so it says the area is a TS we're working backwards a little bit here so we're gonna use half a B sine C again so half a B sine C but this time it gives us the answer so the answer is that half a B sine C has to equal 80 so let's just plug in all the numbers here again C being my angle and then a and B being my other lengths so if I was to type this into the formula we'd have 1/2 times 11 times 16 times sine C and it would equal 80 now I can't actually work that out but what I can do is I can work this bit out so if I do 1/2 times 11 times 16 we get the answer 88 so we have 88 times sine C equals 80 so we can rearrange this look we can get both sides and divide them by 88 and they'll just give us sine C on its own so sine C equals an 80 divided by 88 is stick as a fraction actually it is a decimal there it comes out as not 0.9 zero and that is actually nine zero recurring there we go I'm gonna leave on my calculator screen and then obviously to finish that off you do the inverse sine again just like we have on these other questions so I'm gonna do that over here left so sign minus 1 of this number here which is actually also a fraction it's 10 over 11 so I minus 1 10 over 11 or that note point 9 0 recurring and we get the answer of 65 0.38 degrees there we go so we can find an angle as well just working backwards if we're given the answer and you can apply that logic there's are lots of different questions where it gives you the answer and you can use your formula backwards just dividing everything over to the other side okay so we've got a vector this question says B is the midpoint of AC so that's this point here is the midpoint so straight away that tells us look because you can see some of the vectors on here that this must also be a and again there are so many different variations of vectors here this use a warn idea of how to approach there's so many different types of questions you can get we could spend an hour to see I was just looking at vectors M is the midpoint of PB so we've got a midpoint here so that's halfway along and then show that NMC is a straight line so straight away I'd like to draw an MC and just think about that so that it's saying is show that that's a straight line we're gonna have a look at some of the vectors on this line and see if we can get a common multiple or a common bracket however you've looked at this before so all I want to have a look to start with is that line n to see I've got three vectors I can look at I've got the full line from n to see we've got from n just to M so I'm go from n to M or I could go from M to C and they're all little parts of this line ok the full line there that's just highlight this the full line the first or not I've written is end to see all the way the second one I've drawn is n to M from there to there and the third will not going on look have a look at it from M to see from there to there now it's up to you which one you choose sometimes some of them are easier than others but the first one there so this n to C is quite simple for us to do now to get from n to C I've got to move backwards up the line this way and then I can go forwards along to see there so that first part of the journey going from let's get rid of that going from n up to a and let's draw this in to go from N to a so I'm just going to write this down is - to be you have to get backwards through that to be sacrament to see first of all it's - to be then I can go from a over to see doing two A's and that's positive so plus 2ei there yes I've got - TB plus it's away now whenever it comes to these vectors here you always just want to look to factorize them so I'm going to move this out of the way I'm just gonna see what does that factorize t now I can factorize it by two and I would get to lots of minus B plus a and also we could write and it's likely from where I could swap the letters around if I want I could write to lots of a minus B now if I can prove that one of these other vectors here that I'm looking at below has this bracket a minus B then they must be on the same straight line because therefore they're going in the same direction so let's have a look at one of them so I can either look at n - M or M - C it doesn't really matter I'm just gonna go with em to see either one here are gonna be just as difficult for us to do without first one there is nice and easy so always make sure you have a go at finding some of the lines now that M letter here is halfway along PB so I'm gonna have to find the vector for P to be and that's gonna be my absolutely key one here P to be because if I can't find the full length of the line I'm definitely not going to find halfway along it so if we look at how do we get from P to B well that's actually quite simple you can go from P up to a and that's - three B's so let's write that down you've got you've got the - B to get to N and then another - to get up to a so - three B's and then to get from A to B you do a plus a moving along that vector there so plus a so that is my vector P to B so there's a very good one to always make sure you get that missing lying whether you've got a halfway point now in order to move halfway along that line I'd have to do half of this vector so to get from P to M or from M to B which is half of that let's go for M to be is that's the quickest way for us to move there M to be we'd have to do half of this vector so two times up by 1/2 I'm just gonna write it as 1/2 lots of minus 3b plus a so half of that now if I expand that I can I can actually add stuff to this as well because that only gets us from M to B if I draw that on that gets us from here to here doing half of that line so you expand that out minus three times a half is minus three halves so - that's over 2b plus half of a so half a and now we can finish that off because now we're at B we can move our final a little bit of the line which is to go over here to see and that's adding an additional a in there okay so that's our last little bit plus a and if we add this all together and let's see what we get so we have minus 3 over 2 B I'm gonna write this over here that's not changing so we've got minus 3 over 2 B and then we've got half a Adonai and that gives us 1 point 5 ale or another 3 over 2a ok 3 over 2a so let's write that down over here we have minus 3 over 2 B plus 3 over 2a so that is our vector there to get from M to see and what I need to do is just factorize this one if you have a look they both divide by 3 over 2 so I can take this 3 over 2 out the bracket and if we divide them both by 3 over 2 we get minus B plus hey there we go and I rewrote that a different way as well I could write that as 3 over 2 brackets a minus B the other way around and there we go we've got that matching bracket there in both of them okay so both share that common vector there that common movement between the between the between the points on the diagram nm and C again you could have found n - M as well we could do the same thing and actually because we found this half this one just here we can actually look just think if I was moving from n to M so ends when we would do plus B and then half of that vector just there and if we did do that let's just do it over to the side so this is from n - M not that we didn't need it this one we only ever need to so from n to M I could do plus B or just be and then add to that this vector down here that we did in the green down there so add to that minus 3 over 2 B plus 1/2 a and if we add that all together they're minus 1.5 plus 1 gives us minus 1/2 so we'd have minus 1/2 B that's 1/2 a and again we could factorize that by 1/2 so we get 1/2 minus B plus a and again we get that minus B plus a there which we can write it in a different way so we could have done any three of them all three of them eventually give us this same bracket here okay so we have the same multiple it means it's on the same line and there's a bit of vectors for you and the last question here is just involving some circle sectors but also involving a little bit of trigonometry as well it says that this is an arc of a circle Center 0 it wants us to work out this shaded region so if we can work out the full sector we can take away this unshaded triangle so to work out the area of a sector we would have to do PI R squared they're not forgetting the area of a circle formula PI squared and multiply that by whatever fraction of the circle this is know says 35 degrees over there so it's 35 over 360 so if we plug those numbers in it'll give the area of the sector so we've got pi times 80 squared centimeters multiplied by 35 every 36 thing type that on the calculator let's see what we get pi times 80 squared x 35 over 360 and we get the answer 1954 0.7 it round us around at the two more places 77 obviously be careful what the question says again that's meter squared then we can work out the area of the triangle and obviously not forgetting just because it's not just a triangle in its own we can still use our half a DB sine C so we can use half a B sine C to get this one and if we plug the numbers in 35 there is our C and because it's a circle sector a and B here are the same because it's the radius of the circle so we can do 1/2 times 80 times 80 times sine 35 there we go if we type all that in not 0.5 times 80 times 80 times sine 35 we'll get the area of that triangle there which is 1835 0.44 meter squared and then to finish that off we can do the biggest shape the sector subtract the area of the triangle so one nine five four point seven seven take away this one eight three five point four four and that leaves us with an overall area here it's working out one nine five four point seven seven takeaway answer and we get a hundred and nineteen point three to five meters squared and there we go and obviously just rounding that however you'd be asked in the question so there's quite a large selection of geometry questions there wouldn't on the geometry [Music] [Music]