A Satisfying Geometry Problem

Added: 2026-05-15

[00:00:00]Take a look at this one. We've got a big right triangle ABC.

[00:00:08]And tacked up in the top corner is a smaller right triangle A E D.

[00:00:16]Our goal is to find this little slanted segment over here on the right. D C.

[00:00:24]Let's call that length X.

[00:00:28]Now let's gather the information they give us.

[00:00:32]The horizontal segment E D is 9.

[00:00:37]The lower vertical piece EB is four.

[00:00:43]And here's the key clue in the whole diagram.

[00:00:48]The top vertical segment AE is exactly the same length as the entire bottom base BC. So instead of giving them different variables, let's just call both of them Y.

[00:01:07]Now the moment you see this setup, similarity should start flashing in your head. The small triangle A E D and the large triangle A B C both share the exact same angle at A.

[00:01:29]And both triangles also have perfect 90° corners.

[00:01:35]That means the two triangles are similar.

[00:01:40]And once triangles are similar, their corresponding sides must have the same ratio. So let's match the vertical side with the horizontal side in both triangles.

[00:01:56]For the small triangle, the vertical side is y and the horizontal side is 9.

[00:02:06]For the large triangle, the full vertical side is y + 4 and the full horizontal side is y.

[00:02:17]So we get y over 9 = the quantity y + 4 / y.

[00:02:28]Now let's cross multiply.

[00:02:31]y^2 = 9 * the quantity y + 4.

[00:02:39]Expand the right side.

[00:02:42]y^2 = 9 y + 36. Bring everything to one side.

[00:02:51]y^2 - 9 y - 36 = 0.

[00:03:00]Now we just factor it. We need two numbers that multiply to - 36 but add up to -9.

[00:03:14]Those numbers are -12 and pos3.

[00:03:20]So the equation becomes y - 12 * y + 3 = z.

[00:03:32]That gives us two possible answers. y = pos12 or -3.

[00:03:41]But a length cannot be negative. So we throw out -3.

[00:03:47]That means y = 12.

[00:03:54]Now everything becomes really clean.

[00:03:59]Look at the small triangle first. Its vertical side is 12 and its horizontal side is 9.

[00:04:10]That's a classic 345 triangle scaled up by three.

[00:04:20]So the hypotenuse a must be 15.

[00:04:26]Now move to the large triangle.

[00:04:29]Its total height is 12 + 4 which is 16 and its base is 12.

[00:04:39]That's another 345 triangle. This time scaled up by four.

[00:04:50]So the full hypotenuse AC is 20.

[00:04:56]And now we're basically done. The whole slanted side is 20. The upper portion a d is 15.

[00:05:08]So the remaining piece DC c which is our x is simply 20 minus 15 and that gives us x= 5.

[00:05:23]Pretty satisfying problem honestly. Once the similar triangles appear the rest of the geometry almost solves itself.