In a triangle the length of the side opposite the angle which measures 45° is 8 cm, what is the length of the side opposite to the angle which measures 90°?

According to the question,

AB = 8 cm.

∠ABC = 90°

$\therefore \sin 45^{\circ}=\frac{\mathrm{AB}}{\mathrm{AC}}$

$\Rightarrow \frac{1}{\sqrt{2}}=\frac{8}{\mathrm{AC}}$

$\Rightarrow \mathrm{AC}=8 \sqrt{2} \mathrm{~cm} .$