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

Question

In the following picture, $XY = YZ$, $\angle a = 40^\circ$, and the length opposite is $5$.

The problem asks to compute $XY$

We immediately dropped the dotted perpendicular and get two right triangles. The length of the bottom is 2.5 in each right triangle, and therefore $a/2 = 20^\circ$

$\sin 20 = 2.5 / XY$ or, $XY = 2.5 / \sin 20 \approx 7.3$

Their answer uses $5 / a = XY / 70$

$5 / 40 = XY / 70$ $XY = 350/40 = 8.75$

I would have thought both these answers should be the same, where did we go wrong?

Answer 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.

Answer 2

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

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

Answer 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.

Answer 4

As others noted, the rule would be $\frac{5}{\sin a} = \frac{XY}{\sin 70°}$ and from there some editor multiplied both sides by $\sin$ and omitted the $°$ from the angle. Which is practical joke level mathematics like $\frac{16}{64}=\frac{1\not 6}{\not 64}=\frac14$. Except that in this case the result is wrong. Which means that the person responsible for doing the final substantial corrections to the text (whether its author or not) was unqualified to do so. That does not bode well for the rest of the book.