Certainly! Let's complete the proof step-by-step:
Given:
Isosceles right triangle \( XYZ \) is a \( 45^{\circ}-45^{\circ}-90^{\circ} \) triangle.
To Prove:
In a \( 45^{\circ}-45^{\circ}-90^{\circ} \) triangle, the hypotenuse (\( c \)) is \( \sqrt{2} \) times the length of each leg (\( a \)).
1. State the Pythagorean Theorem:
Since triangle \( XYZ \) is a right triangle, the side lengths must satisfy the Pythagorean theorem:
[tex]\[
a^2 + b^2 = c^2
\][/tex]
2. Account for Isosceles Triangle Properties:
In an isosceles right triangle, the legs \( a \) and \( b \) are of equal length:
[tex]\[
a = b
\][/tex]
3. Substitute \( b \) with \( a \) in the Pythagorean Theorem:
[tex]\[
a^2 + a^2 = c^2
\][/tex]
4. Combine Like Terms:
[tex]\[
2a^2 = c^2
\][/tex]
5. Divide Both Sides by 2:
To isolate \( a^2 \):
[tex]\[
a^2 = \frac{c^2}{2}
\][/tex]
6. Solve for \( c \):
To express \( c \) in terms of \( a \), take the principal square root of both sides:
[tex]\[
c = \sqrt{2a^2}
\][/tex]
7. Simplify the Expression:
[tex]\[
c = \sqrt{2} \cdot a
\][/tex]
Conclusion:
The length of the hypotenuse [tex]\( c \)[/tex] is [tex]\( \sqrt{2} \)[/tex] times the length of each leg [tex]\( a \)[/tex] in a [tex]\( 45^{\circ}-45^{\circ}-90^{\circ} \)[/tex] triangle. This completes the proof.
