Question

1. in the right triangle with a 45° angle, one leg is $2\sqrt{10}$, find the lengths of x and y.

Explanation:

Step1: Identify the triangle type

This is a right - isosceles triangle (one angle is \(90^{\circ}\), another is \(45^{\circ}\), so the third is also \(45^{\circ}\)), so the two legs are equal, i.e., \(x = 2\sqrt{10}\).

Step2: Use the Pythagorean theorem to find \(y\)

For a right - triangle, \(y^{2}=x^{2}+(2\sqrt{10})^{2}\). Since \(x = 2\sqrt{10}\), we substitute:
\(y^{2}=(2\sqrt{10})^{2}+(2\sqrt{10})^{2}\)
\(y^{2}=4\times10 + 4\times10\)
\(y^{2}=40 + 40=80\)
Then \(y=\sqrt{80}=4\sqrt{5}\) (we take the positive root because the length of a side is positive)

Answer:

\(x = 2\sqrt{10}\), \(y = 4\sqrt{5}\)