Let's solve the problem step-by-step:
1. Given: An isosceles right triangle \(XYZ\) with angles \(45^\circ\), \(45^\circ\), and \(90^\circ\).
2. Objective: To prove that in a \(45^\circ\)-\(45^\circ\)-\(90^\circ\) (isosceles right) triangle, the hypotenuse is \(\sqrt{2}\) times the length of each leg.
3. Pythagorean Theorem Application: In any right triangle, the side lengths must satisfy the Pythagorean Theorem: \( a^2 + b^2 = c^2 \).
Since triangle \(XYZ\) is isosceles:
- Let \(a\) be the length of each leg (since both legs are of equal length).
- Let \(c\) be the length of the hypotenuse.
Given the isosceles property, we substitute \(a\) for both legs:
[tex]\[ a^2 + a^2 = c^2 \][/tex]
4. Combine Like Terms: Combine the terms on the left side:
[tex]\[ 2a^2 = c^2 \][/tex]
5. Determine the principal square root of both sides:
To isolate \(c\), we need to solve for \(c\) by taking the principal square root of both sides:
[tex]\[ c = \sqrt{2a^2} \][/tex]
6. Simplify the Square Root: Since \(\sqrt{2a^2} = \sqrt{2} \cdot \sqrt{a^2}\), we get:
[tex]\[ c = \sqrt{2} \cdot a \][/tex]
Thus, we have proved that the length of the hypotenuse \(c\) is \(\sqrt{2}\) times the length of each leg \(a\).
Therefore, in a [tex]\(45^\circ\)[/tex]-[tex]\(45^\circ\)[/tex]-[tex]\(90^\circ\)[/tex] triangle, the hypotenuse is [tex]\(\sqrt{2}\)[/tex] times the length of each leg, as desired.
