Question

x =
y =
(right triangle with a 45° angle, hypotenuse 2, one leg x, the other leg y)

Explanation:

Step1: Identify triangle type

The triangle is a right - isosceles triangle (one angle is \(90^{\circ}\), another is \(45^{\circ}\), so the third angle is also \(45^{\circ}\)). In a right - isosceles triangle, the two legs are equal, and the hypotenuse \(h\) and leg \(l\) are related by \(h = l\sqrt{2}\). Let the legs be \(x\) and \(y\), and the hypotenuse be \(2\).

Step2: Find the length of \(x\) (and \(y\) since \(x = y\))

We know that for a right - isosceles triangle, \(\cos(45^{\circ})=\frac{x}{2}\) (adjacent/hypotenuse). Since \(\cos(45^{\circ})=\frac{\sqrt{2}}{2}\), we have \(\frac{\sqrt{2}}{2}=\frac{x}{2}\). Cross - multiplying gives \(x = \sqrt{2}\). Also, since it is an isosceles right triangle, \(y=x=\sqrt{2}\).

Answer:

\(x = \sqrt{2}\), \(y=\sqrt{2}\)