Question

find the missing side length. all answers must be in radical form.
(there is a right triangle with one angle 45°, hypotenuse (5sqrt{2}), legs (a) and (b). below the triangle is a table with 45, 45, 90 in the first row, x, x, (xsqrt{2}) in the second row, and three dashed boxes in the third row. at the bottom, there are multiple - choice options: (2sqrt{5}), (2sqrt{10}), 5, (5sqrt{2}), 10, (10sqrt{2}))

Explanation:

Step1: Identify triangle type

This is a 45-45-90 triangle, so legs are equal (\(a = b\)) and hypotenuse \(= x\sqrt{2}\), where \(x\) is leg length.

Step2: Relate hypotenuse to leg

Given hypotenuse \(= 5\sqrt{2}\), set \(x\sqrt{2}=5\sqrt{2}\).

Step3: Solve for \(x\) (leg length)

Divide both sides by \(\sqrt{2}\): \(x = \frac{5\sqrt{2}}{\sqrt{2}} = 5\). So \(a = b = 5\).

Answer:

\(a = 5\), \(b = 5\) (choosing the option "5" as the missing side length for \(a\) and \(b\))