[Music] in this video we're going to look at similar triangles we're going to look at two types of problem one's where the triangles are connected like this one and ones where they overlap like this one here before we do this we're going to take a closer look at what it means for two triangles to be similar take these two triangles here if these two triangles are similar there must be a constant scale for out of enlargement from one to the other so if we match up the corresponding sides on the base here we have 12 and 24 on the left we have 5 and 10 for the height and the hypotenuse of these are 13 and 26 to get from 12 to 24 we'd multiply by two this is the same as going from 5 to 10 on the height and also for the hypotenuse from 13 to 26 since this is always the same number these shapes must be similar it works in the other direction as well so if we go from 24 to 12 we would multiply by 1/2 and and this is the same for the other sides as well so we know these two shapes are similar there are some other observations we can make we can find that scale factor of enlargement by dividing the pairs of corresponding sides so if we take the purple sides 24 and 12 and divide them you'll see we get two this works for the other pairs of sides as well so 10id 5 that's also 2 and 26id 13 that's once again two but it also works the other way around so if we take those fractions and do the reciprocal like this then these will always give you the same value as well which this time is 1/2 in these fractions here we divided one side from one of the triangles by the corresponding side on the other triangle but we can actually just divide sides within the same triangle as well so if we take the green side in the small triangle and divide it by the purple side we get 5 over2 and then if we do the same thing in the larger triangle the green side divid by the purple side that's 10id 24 these two fractions actually give the same value and it will be the same for any of the pairs of sides as long as you're consistent in what you do for instance if I take the blue side and divide it by the green side and then do the same on the second triangle blue divide by Green I get the same number once again these two fractions give the same value or I could have done the purple side divide by the blue side so we get 12 over 13 on this one and on the other one 24/ by 26 and these also give the same value and so do their reciprocals so if we take all of these fractions and take the reciprocal of them those will also be vehicle as well so there's actually lots and lots of equivalent fractions that you can write down using similar triangles this is going to be helpful in solving some of the more complicated questions in this video now let's go back to the original type of question the first one we said was when we have two triangles that are connected and notice we have a parallel line at the top and at the bottom the first thing to notice here is that this angle here is the same as the one below it because these are vertically opposite angles then if we draw in this angle here and draw on some lines like this we find that the green angle is the same as the green angle down here and we call these alternate angles you can see we've got a zed shape there the same idea works for this angle as well so if we draw in a zed shape here this time we find that this blue angle is the same as the blue angle at the bottom down here once again due to alternate angles in the previous video on similar shapes we learned that when the angles are all the same the shapes must be similar it's even more obvious if I move this triangle down to the bottom here and flip it round you can see the blue angles on the left are the same so are the red ones at the top and the green ones on the right but I'd also need to move down these sides like this notice how the sides have moved to a different position this time if I put the original diagram back you can see the eight was on the right hand side but now it's on the left hand side the 10 was on the left and now it's on the right and the 13 was on the top and now it's on the bottom so let's go ahead and try and find these missing values now that we've put this information on so to get from 10 to 20 we multiply by two so we must multiply the 13 by two on the bottom to get to Y and 13 * 2 is 26 then of course we must multiply the 8 by two to get to the x value and 8 * 2 is 16 now this uses the technique that we did in the previous video on similar shapes but I'm also going to show you this by using fractions you don't need to use this approach to solve these ones but it's going to help us to practice it especially for the questions later in this video so what we're going to do is pair up the corresponding sides so on the left here we have X and 8 and on the right we have 20 and 10 we're going to use these pairs of sides to find the value of X so if we divide the green ones by doing xide by 8 this must be the same as when we divide the blue ones so 20 over 10 so we form an equation like this we can solve this equation by multiplying both sides by eight if you multiply the left side by 8 the 8 will cancel and then you multiply the right side by 8 we get this 20 * 8 on the top there is 160 so we have 160 over 10 which was 16 and we knew that was the answer to the question because we worked it out before so so X is 16 and let's do a similar idea to work out y so y pairs up with the 13 on the bottom here so if we divide the purple Sid we get y over 13 and this must be equal to any of the other pairs divided I'm going to go for the blue pair again so 20 over 10 here we just multiply both sides by 13 and we get 20 lots of 13 over 10 20 * 13 is 260 and divide this by 10 and you get 26 which again we knew this was the answer from before now let's take a look at how a question like this could be worded so if we take some triangles like this and we're told to work out the values of X and Y notice they haven't even told us that these two triangles are similar they don't need to because they've marked on the parallel lines so we should already know that information so let's try with X first you can see that X matches up with a 15 on the bottom then if we remember that if we flip that triangle from the top upside down the8 is actually going to match up with a 12 so even though the eight is on the right it matches up with the 12 on the left left and the four matches up with the Y so let's divide the purple sides so x / 15 and this is going to be equal to one of the other pairs divided now since we don't know why I'm going to avoid the blue pair and go for the green one so the green one would be eight IDE by 12 then we can just solve this like we did for the previous two we just multiply both sides by 15 so on the left we get X and on the right we get 8 over2 multiplied 15 which we could combine to one fraction like this 8 * 15 is 120 so we get x = 120 / 12 and 120 / 12 is just 10 so we found the value of x it's 10 one thing that's really important is you need to be consistent in the way you set up the fractions at the start of this question the X here was from the smaller triangle the eight was also from the smaller triangle the 15 was from the larger triangle and so was the 12 it's important that you're consistent in having the small on the top and the large on the bottom or of course you could have the large on the top and the small on the bottom it doesn't matter which way around you do it as long as you're consistent I put the small on the top here because we were trying to find X so let's replace the X with a 10 and let's go and find y so this time we're going to divide the blue pair so y over 4 and this is going to be equal to one of the other pairs I'm going to go for the green one once again but this time it will be 12 over 8 notice y was on the larger triangle so 12 must also be on the larger triangle and on the bottom we've got four and eight which are from the smaller triangle to solve this then we just multiply both sides by four on the left we'd have y and on the right 12 8 multiplied by 4 we can combine that to one fraction so it's 12 * 4 over 8 12 fours are just 48 so it's 48 over 8 and 48 / by 8 is 6 so the value of y is 6 now let's have a look at the other type of problem so a question that looks like this this time the triangles are both there but they're overlapping each other we we can show that they're similar Again by looking at their angles so if we draw in these lines here then this green angle on the left must be the same as this green angle up here this is because they're corresponding angles the same idea works on the right side so if we draw in this line and then these two this angle here that's blue must be the same as this one here that's blue once again because they're corresponding angles and then finally we have the angle at the top here well that's actually the same angle for both of the triangles so if I separate them off like this we've got the small triangle and then the large triangle you can see all of the angles match again so they must be similar shapes but this one can be a bit more tricky to do on the top triangle I actually have lengths for all of the sides so I've got the 15 on the bottom the 18 on the left and the 12 on the right on the larger triangle though it's a bit more difficult I can see I have 25 on the base so I can mark that on but the left and right side can be tricky so for the left side here it's the whole length from here to here so we have a y and an 18 so the total length must be y + 18 it's similar on the right hand side so we need to go all the way from here to here which we have X and 12 so the total length is x + 12 now that we form these triangles we're ready to find the missing values of X and Y to do this we're going to divide corresponding pairs of sides again so I'm going to start by trying to find X so I can see I've got X on the larger triangle in the x + 12 side so x + 12 and I'm going to divide that by the side that matches on the smaller shape which is 12 so I have x + 12 over 12 then this is going to be equal to a second fraction and I need to pick another pair of sides now I'm not going to pick the sides on the left because they have a y in them if I pick the base of the triangle then I can see I have both of those sides I've got 15 and 25 I need to be consistent and since I started with the larger triangle on the last fraction I'll do the same here so it's 25 on the top and 15 on the bottom now we can go ahead and solve this so for this one we would multiply both sides by 12 that would give us x + 12 on the left and on the right we'd have 25 multiplied by 12 all over 15 we could also multiply both sides by 15 now so if we multiply the left side by 15 we need to use a bracket so we'd have 15 lots of x + 12 and on the right hand side the 15s will no cancel so it's just 25 * 12 on the left I would now expand out this bracket so 15 lots of X is 15 x and 15 lots of 12 is 180 on the right side I need to do 25 multip by 12 which is 300 this is now just a nice two-step equation to solve we can subtract 180 from both sides so we get 15x = 120 and then divide both sides by 15 we get x = 8 so we found the value of x it's 8 cm now let's do the same idea and work out why so we'll start with the y + 18 and divide it by the side that matches on the other triangle which is 18 and this will be equal to another fraction and it makes sense to choose the sides 25 and 15 again so 25 over 15 you can see I've been consistent again and I have the larger triangle on the top and the smaller triangle on the bottom of those fractions we'll solve this in the same sort of way so we'll multiply both sides by 18 on the left this will give us y + 18 and on the right 25 * by 18 all over 15 then multiply both sides by 15 so we get 15 lots of y + 18 and on the right the 15 cancel so it's just 25 * 18 if we expand out the brackets here we get 15 lots of y That's Just 15 Y and 15 * 18 is 270 on the right 25 lots of 18 is 450 once again we just have a two-step equation to solve if you subtract 270 from both sides we get 15 y = 180 and then divide both sides by 15 and we'll get y = 12 now let's try a second example of this this a little bit more difficult so this time we've got this diagram and we're just trying to find X so if we separate these off into two triangles we have a small triangle that looks like this and we can see the side on the left there at the top is X and on the right we have a nine then if we do the big triangle here the side on the top left is actually all the way from here to here so we need to do x + 5 and on the right this time is 12 now that we have these two triangles we can go ahead and find X so we need to divide the sides that match so I'm going to divide x + 5 on the big triangle by X on the small triangle and this will equal another fraction and since I started with the larger one I'll do 12 over 9 this one's a little bit different in its structure but we use the same idea to solve it let's multiply both sides by X this time on the left this will cancel the X on the bottom so I get x + 5 and on the right I'll get 12 lots of X over 9 then we multiply both sides by 9 on the left hand side this will give us 9 lots of x + 5 and on the right the N9 will cancel so we just have 12 lots of X expanding the bracket on the left gives us 9x + 45 and on the right 12 lots of X is just 12 x to solve this equation we' subtract 9x from both sides if you subtract it from the left it will cancel so we've just got 45 and if you subtract 9x from 12x you get 3x then finally divide both sides by three and you find that X is equal to 15 now I want to look at one more of these questions but there's going to be some ratio involved so if we take this diagram here and notice we actually have labels for the sides this time this is quite common in exams especially for Ed Exel and this time they're going to tell us that the ratio of a d to be is 5 to 4 and we're going to be tasked with finding the length of De now at first glance it looks like we don't have very much information on this diagram we only have the 15 cm so it might feel like we don't have enough to answer the question but we do we're going to start with this ratio here A D to be being at 5: 4 now the easiest way to think about this is just to pretend that a d is 5 cm and that be is 4 cm even if it doesn't look like they're that long in the diagram it's still going to work for the question so I'm going to label be as four and AD as five from here we just proceed with the question as normal it's important that you understand that those aren't actually that length it's just their ratio but for this question it will help us solve the problem finally since we're trying to find de I'm going to label that as X so what we'll do now is separate them off into the two triangles so we have the small triangle that looks like this that's triangle EBC and for this triangle we know some of its lengths from E to C is 15 and E to B is four or at least we're pretending it's four and for the large triangle which is triangle da we know the height on the left from a to d is 5 or at least we're pretending it's five and we also know the length all the way along the diagonal there is x + 15 now we just go ahead and set up some fractions again so I'm going to start with x + 15 and divide it by the same side on the other triangle which is 15 then I have my second fraction that this is equal to and it will be 5 over 4 now we just solve this equation like we have done all of the previous ones multiply both sides by 15 that's x + 15 on the left and on the right 5 * 15 all over 4 then multiply both sides by four on the left we've got four lots of x + 15 and on the right the four will cancel so 5 lots of 15 expand out the bracket on the left we've got 4x + 60 and on the right 5 * 15 is 75 now solve this two-step equation by subtracting 60 from both sides 4X will be 15 and then finally divide both sides by four and we'll find that X is 3.75 thank you for watching this video I hope you found it useful check out the one I think you should watch next subscribe so you don't miss out on future videos and go and try the exam questions on this topic that are in this video's description