To prove that in a \(45^{\circ}-45^{\circ}-90^{\circ}\) triangle, the hypotenuse is \(\sqrt{2}\) times the length of each leg, follow these steps:
1. Start with the Given Information:
A \(45^{\circ}-45^{\circ}-90^{\circ}\) triangle is an isosceles right triangle. Therefore, the two legs (let's denote them as \(a\)) are equal in length, and the hypotenuse (denoted as \(c\)) is the side opposite the right angle.
2. Apply the Pythagorean Theorem:
For any right triangle, the Pythagorean theorem states:
[tex]\[
a^2 + b^2 = c^2
\][/tex]
Since this is an isosceles right triangle, both legs are equal (\(a = b\)). Thus, the theorem simplifies to:
[tex]\[
a^2 + a^2 = c^2
\][/tex]
3. Combine Like Terms:
Combining the terms on the left side of the equation, we get:
[tex]\[
2a^2 = c^2
\][/tex]
4. Isolate \(a^2\):
To simplify, we divide both sides of the equation by 2:
[tex]\[
a^2 = \frac{c^2}{2}
\][/tex]
5. Determine the Principal Square Root of Both Sides:
Now, take the square root of both sides to solve for \(a\):
[tex]\[
a = \sqrt{\frac{c^2}{2}}
\][/tex]
Given that \(c^2\) is the square of hypotenuse, the square root of \(c^2\) is \(c\). Thus we can rewrite \(a\) as:
[tex]\[
a = \frac{c}{\sqrt{2}}
\][/tex]
6. Multiply both sides by \(\sqrt{2}\) to isolate \(c\):
To solve for \(c\), multiply both sides of the equation by \(\sqrt{2}\):
[tex]\[
a \sqrt{2} = c
\][/tex]
Thus, the length of the hypotenuse [tex]\(c\)[/tex] is [tex]\(\sqrt{2}\)[/tex] times the length of each leg [tex]\(a\)[/tex]. This completes our proof.
