Finding missing length using similar triangles

Question

How do you solve the above problem? The only solution I can come up with is to perform pythagoras on triangle EDC and find the length of DC. Then I would use similar triangles to find the length of AC, before subtracting the length of DC from this answer.

Yet how would you solve this problem efficiently in an exam setting that does not permit a calculator. Is it possible? My method requires taking the square root of a large number.

Answer 1

The intended trick here is to note that $\frac{EC}{DE}=\frac54$, i.e. the triangles are 3-4-5 right triangles. Then $DC=1905\cdot\frac35=1143$, $AC=1016\cdot\frac54=1270$ and the answer is A.

Answer 2

You know $ED$ and $EC$, then, with Pythagoreon theorem, you can find $DC$. For $AC$, since $ABC$ and $EDC$ are similar, you have that $AC=\frac{AB}{DE}\cdot EC$. I let you conclude.