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10 Jul 2023, 03:08
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IanStewart
IanStewart
If I understand correctly, the two triangles formed by the perpendicular line are similar to the original triangle and to each other.
Since they are all similar, why is using the triangle PSR in the ratio producing a different answer?
I'm getting d/4 = 4/5 --> d = 16/5 (in fact, I’m also getting 9/5 depending on the triangles and sides I use)
I’ve double checked to make sure the orientations of the triangles are the same, so I'm really not sure where I'm going wrong.
Thanks in advance!
10 Jul 2023, 08:25
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achloes
I think you've just not matched up the sides of the triangles correctly, which is a bit tricky in this question, because the triangles are rotated. To correctly match up sides in two similar triangles, you always want to look at the opposite angles. In the 3-4-5 triangle, the hypotenuse, opposite the 90 degree angle, is 5. In triangle PRS, the hypotenuse is 4. So triangle PRS is just 4/5 as big as the 3-4-5 triangle, and its three sides are 12/5, 16/5, and 4. The only remaining question is: is the length of d 12/5, or is it 16/5? Again, we look at opposite angles: side d is opposite angle S in triangle PRS. In the big 3-4-5 triangle, the short side of length 3 is opposite angle S. So those two sides match, and d = (4/5)(3) = 12/5.
You might be thinking about similarity in a slightly different (but equivalent and perfectly good) way. The ratio of d (opposite angle S) to 4 (opposite the right angle) must equal the ratio of 3 (opposite angle S) to 5 (opposite the right angle), so d/4 = 3/5, and d = 12/5.
If you want to confirm you can use similarity in this way, you could now try finding d using triangle PQR. If you get 12/5 as the answer, you've done everything correctly. And if you can apply similarity correctly here, this is about as hard as it can get on the GMAT, so you'll be fine for anything you see on the test (that said, the area solution to this particular question is the fastest solution by far).
20 Aug 2025, 01:54
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