Australian AMC 2022 - Senior Question 23

Question 23

A semicircle is inscribed in a right-angled isosceles triangle and a square is inscribed in the semicircle as shown. What is the ratio of the area of the square to the area of the triangle?

Question illustration

$1:3$
$1:\sqrt2$
$2:5$
$1:2\sqrt2$
$3:8$