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# Solving similar triangles: same side plays different roles

CCSS.Math:

## Video transcript

in this problem we're asked to figure out the length of BC we have a bunch of triangles here and some lengths of sides and a couple of right angles and so maybe we can establish similarity between some of the triangles there's actually three different triangles that I can see here this triangle this triangle and this larger triangle if we can establish some similarity here maybe we can use ratios between sides somehow to figure out what BC is so when you look at it you have a right angle right over here yeah so in triangle BDC you have one right angle and triangle ABC you have another right angle if we can share if we can show that they have another angle or another corresponding set of angles that are congruent to each other then we can show that they're similar and actually both of those triangles both BD c and ABC both share this angle right over here so if they share that angle then they definitely share two angles so they both share that angle right over there let me do that in a different color just for just to make it different than those right angles they both share that angle there and so we know that two triangles that have at least two of their angles are have at least two congruent angles they are going to be similar triangles so we know we know that triangle I'll write this triangle ABC ABC we went from the unlabeled angle to the right yellow right angle to the orange angle so let me write it this way went from the unlabeled angle right over here to the orange angle or sorry to the yellow angle I'm having trouble with colors to the orange angle ABC and we want to do this very carefully here because the same points or the same vertices might not say play the same role in both triangles so we want to make sure we're getting the similarity right white vertex to the the 90 degree angle vertex to the orange vertex that is going to be similar to to triangle so which is the one that is neither right angle so we're looking at the smaller triangle right over here which is the one that is neither a right angle or the orange angle what's going to be vertex B vertex being at the right angle when you think about the larger triangle but we haven't thought about just that little right over there so we start at vertex B then we're going to go to the right angle the right angle is vertex D vertex D and then we go to vertex C which is in orange so so we have shown that they are similar and now that we're sin that we know that they are similar we can attempt to take ratios between the sides and so let's think about it we know what the length of AC is AC is going to be AC is going to be equal to 8 6 plus 2 so we know that AC AC what's the corresponding side on this triangle right over here so you can literally look at the letters A and C is going to court is going to correspond to BC the first and the third first and the third ac is going to correspond to BC and what is and so this is interesting because we're already involving BC and so what is going to correspond to and then if we look at BC on the larger triangle so if we look at BC on the larger triangle BC is going to correspond to what on the smaller triangle it's going to correspond to DC and it's good because we know what AC is and we know what DC is until we can solve for BC so I want to take one more step to show you what we just did here because BC is playing two different roles on this first statement right over here we're thinking of BC BC corresponds BC on our smaller triangle corresponds to AC on our larger triangle and then in art then the second statement BC on our larger triangle corresponds to DC on our smaller triangle so in both of these cases so these are our larger triangles and then these are this is from the smaller triangle right over here corresponding sides and this is a cool problem because BC plays two different roles in two in both triangles but now we have enough information to solve for BC we know that AC is equal to 9 we know that a sorry AC is equal to 8 AC is equal to 8 6 plus 2 is 8 and we know that DC is equal to 2 that's given and now we can cross multiply 8 times 2 is 16 is equal to BC times BC is equal to BC squared and so BC is going to be equal to the principal root of 16 which is 4 bc is equal to 4 bc is equal to 4 and we're done and the hardest part about this problem is just realizing that bc plays two different roles and just keeping your head straight is just keeping your head straight on those two different roles and just to make it clear let me actually draw these two triangles separately so if i drew ABC separately it would look like this it would look like this so this is my triangle a B C and then this is a right angle this is our orange angle we know that the length of this side right over here is 8 and we know that the length of this side what we figured out through this problem is 4 then if we wanted to draw B DC we would draw it like this so BD c pd c looks like this so this is b d see that's a little bit easier to visualize because we've already this is our right angle this is our orange angle and this is 4 and this right over here is 2 and i did it this way to show you that you kind of have to flip this triangle over and rotate it just to have kind of a similar orientation and then it might make it look a little bit clearer so if you found this part confusing i encourage you to try to flip and rotate BD c in such a way that it seems to look a lot like ABC and then this ratio should hopefully make a lot more sense