ACT 17: Given Similar Triangles and Congruent Lengths, Solve for Missing Length

Added: 2026-05-20

[00:00:00]Hello and welcome. So we're going to be answering  a question which is, in the figure below we have two triangles. Triangle ABC is similar to triangle  DEF. A little squiggle means similar right there.

[00:00:12]Okay. And then we're told that D and E are lying  on AB, right? And that just means that you don't assume that D and E are on AB but we're actually  told that while it looks like it, they are on the line. They actually are. Okay. Now we're told  that given B to E, which is that segment right there, is congruent to A to D, meaning that whatever this  segment is, this segment has to be the exact same thing. Okay, those two segments are congruent to  each other. And the other lengths in the figure below. So we're given 29 and 145. Okay, what  is the length of A to D? So that length, end goal.

[00:00:55]So when we're looking at this one, there's  a couple ways we could set this up. Similar triangles mean what it is by definition is  that you have between your triangles this proportionality, meaning that if you multiply  29 by some number in order to get to 145, which is going to be whatever 145 divided by  29 is. Okay, that number is going to be the same proportion where you would have to multiply  13 by that same number in order to get to this whole side, right? Whatever this side was, you'd  have to multiply it by that same number to get to that side. And that is what it means to have  proportions, equal proportions on all sides.

[00:01:32]In other words, similar triangles. So there's a  couple ways we could do this. All right, we could um write it like an actual equation for this  thing, because we know that if we have 145 over 29, that proportion has to be the same thing as uh, whatever this entire side is over 13. Because you have two triangles right here. Here's your  small triangle versus here is the big triangle, right? So you're always going, here we did 145  over 29 which means we're doing large to small.

[00:02:09]Okay, which means we should say large value over  small value. Keep it consistent on the other side of your equal sign for your other proportion.  So that's all right. Well what's the entire side here? Well we know 13 is going to be a part  of it because we can see 13 there. We have three portions of this side: we have portion 1, 13,  and portion 2. Oh we already know that's 13, and we know that because we're told that these  two sections are congruent to each other. Whatever that is, this has to be the same number. Well  what happens if we add all of those together?

[00:02:43]We get 13 plus two x's. Whatever that x value is,  that's the value we actually need to find. Okay, because that's the length of AD. So then you have  an equation. Solve for it. You multiply both sides by 29 because then that removes the 29 in this  side and then we'll have that times 29: 13 plus 2x. You could also multiply by 13 on the same, same step  because that'll wipe out the 13s and leave all of your fractions gone. So then you're just left  with 145 times 13 which is 1885 equals uh 29 to the (13 plus 2x) in parentheses. So we can distribute  that through. So I'd say 29 times 13 is going to leave us 377. Whoops, as my 7 looks a little  weird there. Plus 29 times 2 which is going to be 58x.

[00:03:48]And if we want to solve this equation,  subtract the 377 to the other side.

[00:03:56]So then we have, so let me write that out: 1508  equals 58x. Undo the multiplication by division.

[00:04:08]So from there you would divide both sides by 58  and it leaves us with just the x. So x is then 1508 divided by 58, or in other words 26 is  our remaining side. Okay, and then that means that 26 is your x value which we know is going  to be both of these. We set it out that way.

[00:04:33]Bam. Okay, another way that we could  have done this uh is without setting up an equation at all. We could have instead  said, oh look um if we do 145 divided by 29, right, we get 5 from that. I didn't realize it  simplified up there. Oh, details details, it's fine. Okay, 145 over 29 equals 5. That means that  whatever we have on this side, multiply it by 5, you get the larger triangle. So if we know, let  me just clear a lot of my writing on this, on this thing. If we know that if we multiply the  small side by 5 we get the entire thing, well 13 times 5 is 65. That means that the  entire side, this entire side has to be 65. Because we know that that's the proportion:  multiply by 5 we get to the whole thing. And if we know that that's made up by two equal  portions, all we do is take out the 13 from the 65. We don't care about that. We're trying  to find the little portions. So 65 minus 13, that gives us 52. And that means that  our two x portions must be the 52 and then that means our x has to be from there  26 because that is splitting the 52 equally between two of those sides. And then that adds  up to 65. Okay, that is another way you could have done it. Either/or works. Okay, those are  both the same ways to go about the same problem.

[00:06:11]Different ways to go about the same  problem. Alrighty, thanks for watching.