Question

How do you work this out?

Answer 1

You can find x by using the pythagorean theorem:

$(2x+1)^2+(4x)^2 = (5x)^2$

$4x^2+4x+1+16x^2=25x^2$

$20x^2+4x+1 = 25x^2$

Bring everything to the same side:

$5x^2-4x-1=0$

The solutions to this equation are $-\dfrac{1}{5}$ and 1. We can't accept the negative solutions because it would lead to negative lengths, so we have $x=1$

This implies that the triangle is a 3-4-5 triangle. Now, you can use the law of sines to state

$\dfrac{5}{\sin(90)} = \dfrac{4}{\sin(y)} \iff \sin(y) = \dfrac{4}{5} \iff y = \arcsin\left(\dfrac{4}{5}\right)$