To complete the proof that in a [tex]\(45^{\circ}-45^{\circ}-90^{\circ}\)[/tex] triangle, the hypotenuse is [tex]\(\sqrt{2}\)[/tex] times the length of each leg, we can follow the steps as outlined below:
Given:
- Triangle [tex]\(XYZ\)[/tex] is an isosceles right triangle, also known as a [tex]\(45^{\circ}-45^{\circ}-90^{\circ}\)[/tex] triangle.
- Let the legs of the triangle be [tex]\(a\)[/tex] and [tex]\(a\)[/tex], and the hypotenuse be [tex]\(c\)[/tex].
Step-by-Step Solution:
1. Initial Equation:
We start by using the Pythagorean theorem for the right triangle [tex]\(XYZ\)[/tex]:
[tex]\[
a^2 + b^2 = c^2
\][/tex]
Since [tex]\(XYZ\)[/tex] is an isosceles right triangle, the lengths of the legs are equal, so [tex]\(a = b\)[/tex]. Thus, the equation becomes:
[tex]\[
a^2 + a^2 = c^2
\][/tex]
2. Combine Like Terms:
By combining the terms on the left side of the equation, we get:
[tex]\[
2a^2 = c^2
\][/tex]
3. Taking the Square Root:
To find the length of the hypotenuse [tex]\(c\)[/tex], we need to determine the principal square root of both sides of the equation:
[tex]\[
c = \sqrt{2a^2}
\][/tex]
4. Simplify the Square Root:
Simplifying the square root expression, we can separate the constant factor and the variable:
[tex]\[
c = \sqrt{2} \cdot \sqrt{a^2}
\][/tex]
Since [tex]\(\sqrt{a^2} = a\)[/tex], the equation becomes:
[tex]\[
c = \sqrt{2} \cdot a
\][/tex]
Therefore, we have proven that in a [tex]\(45^{\circ}-45^{\circ}-90^{\circ}\)[/tex] triangle, the length of the hypotenuse [tex]\(c\)[/tex] is [tex]\(\sqrt{2}\)[/tex] times the length of each leg [tex]\(a\)[/tex]. The final answer is:
[tex]\[ c = \sqrt{2} \cdot a \][/tex]
This completes the proof with the final result:
[tex]\[ c = \sqrt{2}a \][/tex]
