Can't understand an ACT practice problem: Triangle appears to be isosceles, why isn't the answer $7.3\sim $ here?

Solution 1:

Get another book.

NOW!

The books answer that $\frac 5{\angle a} = \frac {XY}{\angle 70}$ is ... baseless.

There's no such similarity between triangle sides and the direct measure of angles. It's .... stupid... to think there would be.

But there is a similarity between sides and the SINES of angles.

I.e. The law of sines which would allow us to note:

that $\frac 5{\sin a} = \frac {XY}{\sin 70}$ or $\frac 5{\sin 40}=\frac {XY}{\sin 70}$ so $XY =5*\frac{\sin 70}{\sin 40} \approx 7.3$. Which ... is the same as your answer.

Solution 2:

$$XY=\frac{2.5}{\sin20^{\circ}}=7.3095...$$ Another way:

$$\frac{XY}{\sin70^{\circ}}=\frac{5}{\sin40^{\circ}},$$ which gives the same result: $$XY=\frac{5\sin70^{\circ}}{\sin40^{\circ}}=7.3095...$$

Solution 3:

Their answer uses 5/a=XY/70

This is wrong. There is no such rule. There is a rule of sines. Instead of 5/a=XY/70 there should be 5/sin(a)=XY/sin(70). Then you will get the same result, ~7.3.