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# Using similar & congruent triangles

CCSS.Math:

## Video transcript

so in this problem here we're told that the triangle AC e is isosceles so that's this big triangle right here it's isosceles which means it has two equal sides and we also know from my sausage triangles that the base angles must be equal so these two base angles are going to be equal and this side right over here is going to be equal in length to this side over here we could say AC is going to be equal to C e so we get all of that from this first statement right over there then they give us some more clues or some more information they say C G is equal to 24 so this is C G right over here it has length 24 they tell us BH is equal to DF BH is equal to DF so those two things are going to be congruent they're going to be the same length then they tell us that G F is equal to 12 so this is G F right over here so G F is equal to 12 that's that distance right over there and then finally they tell us that F E is equal to 6 so this is f e and then finally they ask us what is the area of CB h FD so C B H F D so they're asking us for the area they're asking us for the area of this part right over here let me that part and that part right over there that is CB h FD so let's think about how we can do this so we can figure out the area of the larger triangle and then from that we could subtract the areas of these little pieces at the end then we'll be able to figure out this middle area this area that I've shaded and we don't have all the information yet to solve that we don't know we know at the height of or the altitude of this triangle is but we don't know its base if we knew its base we'd say one-half base times height we'd get the area of this triangle and then we'd have to figure out subtract out these areas and we don't have full information there either we don't know this height once we know that height then we could figure out this height but we also don't quite yet know what this length right over here is so let's just take it piece by piece so the first thing we might want to do and you might guess because we've been talking a lot about similarity is making some type of argument about similarity here because there's a bunch of similar triangles for example triangle c c ge shares this angle with triangle DfE they both share this orange angle right here and they both have this right angle right over here so they have two angles in common they are going to be similar by angle angle and you can actually show that there's going to be a third angle in common because these two are parallel lines so we can write that triangle we can write the triangle c GE c ge is similar to triangle DF e to triangle D F E and we know that by angle angle we have one course set of corresponding angles congruent and then this angle is in both triangle so it is a set of corresponding congruent angles right over there and so then once we know that they are similar we can set up the ratio between sides because we have some information about some of the sides so we know that the ratio we know the ratio between DF and this side right over here which is a corresponding side the ratio between DF and CG so the ratio between DF and CG which is 24 is going to be the same thing as the ratio between f e f e which is 6 and G E which is not 12 it's 12 plus 6 it is 18 and then let's see 6 over 18 this is just 1 over 3 you get 3 DF you get 3 3 DF is equal to 24 I just cross multiplied or you could multiply both sides by 24 multiply both sides by 3 you would get this actually you could just multiply both sides times 24 and you'll get 24 over times 1/3 but we'll just do it this way divide both sides by 3 you get DF DF is equal to 8 so we found out that DF is equal to 8 that length right over there and that's useful for us because we know that this length right over here is also this length right over here is also equal to 8 and now what can we do well we can make another than seems like we can make another similarity argument because we have this angle right over here it is congruent to that angle right over there and we also have this angle which is going to be 90 degrees we have a 90 degree angle there and actually said that by itself is actually enough to say that we have two similar triangles we don't even have to show that they have a congruent side here and actually we want to show that these are actually congruent triangles that we're dealing with right over here so we have two angles and actually we could just go straight to that because when we talk about congruence if you have an angle let's convert to another angle another angle let's convert to another angle and then a side that's congruent to another side you are dealing with two congruent triangles so we can right triangle let me write it over here I'll write it in pink triangle a.hb is congruent is congruent to triangle you want to get the corresponding vertices right it can we're trying --gel e FD to triangle e F D and we know that because we angle angle side angle angle side postulate for congruence and if the two triangles are congruent that makes things convenient that means if this side is eight that side is eight we already knew that that's how we establish our congruence but that means if this side is six has length six and the corresponding side on this triangle is also going to have length six so we can write this length right over here is also going to be six now I can I can imagine you can imagine where all of this is going to go but we want to prove to ourselves we want to know we want to know for sure what the area is we don't want to say hey maybe this is the same thing as that let's just actually prove it to ourselves so how do we figure out we've almost figured out the entire base of this triangle but we still haven't figured out the length of Hg well now we can use a similarity argument again because we can see that triangle a BH is actually similar to triangle a CG they both have this angle here and then they both have a right angle they have one a BH has a right angle there AC G has a right angle right over there so you have two angles two corresponding angles are equal to each other you're now dealing with similar triangles so we know that triangle we know that triangle a B H I'll just write it as a HB since I wrote it this way a HB is similar to triangle a G see you want to make sure you get the vertices in the right order a is orange angle G is the right angle and it sees the unlabeled angle this is similar to triangle a G C a G C and what that does for us is that we can use the ratios to figure out what G is equal to so what we could be saying over here well we could say that 8 over 24 BH over its corresponding side of the larger triangle so we say 8 over 24 is equal to 6 is equal to 6 over not Hg but over AG 6 over AG and I think you can see where this is going you have 1/3 1/3 is equal to 6 over AG or we can cross multiply here and we can get a G a G is equal to 18 so this entire length right over is 18 if AG is 18 and a H is 6 then HG is 12 and this is what you might have guessed if you were just trying to guess the answer right over here but now we have proven to ourselves that this base is has length of well we have 18 here and then we have another 18 here so it has a length of 36 so the entire base here is 36 so that is 36 and so now we can figure out the area of this larger of the entire isosceles triangle so the area of AC e is going to be equal to 1/2 times the base which is 36 times 24 and so this is going to be the same thing as this is going to be the same thing as 1/2 times 36 is 18 18 times 24 I'll just do that over here on the top so 18 times 24 8 times 4 is 32 1 times 4 is 4 plus 3 is 7 now we put a 0 here because we're not dealing not with 2 but 20 if 2 times 8 is 16 2 times 1 is 2 plus 1 so it's 360 then you have the two seven plus six is thirteen one plus three is four so the area of a CD of AC e is equal to 432 but we're not done yet this area that we care about is the area of the entire triangle minus minus this area and minus this area right over here so what is the area what is the area of each of these little wedges right over here so it's going to be one half times 8 times 6 so 1/2 times 8 is 4 times 6 so this is going to be 24 right over there and this is going to be another 24 right over there so this is going to be equal to 432 minus 24 minus 24 or minus 48 which is equal to and we could try this to do this in our head if we subtract 32 we're going to get to 400 and then we have to subtract another 16 so if you subtract 10 from for hundreds of 390 so that you get to 300 you get to 384 whatever the units were if these were in meters and this would be meter squared centimeters this would be centimeter squared did I do that right let me go the other way if I add 8 if I had 42 this 24 plus another 8 gets me to 3432 yep we're done