To prove that in a [tex]\(45^{\circ}-45^{\circ}-90^{\circ}\)[/tex] triangle the hypotenuse [tex]\(c\)[/tex] is [tex]\(\sqrt{2}\)[/tex] times the length of each leg [tex]\(a\)[/tex], follow these steps:
1. Understand the Triangle:
- In a [tex]\(45^{\circ}-45^{\circ}-90^{\circ}\)[/tex] triangle, two legs are of equal length, let's call this length [tex]\(a\)[/tex]. The triangle is isosceles, meaning the legs have the same length, and the angles opposite these legs are each [tex]\(45^{\circ}\)[/tex].
2. Apply the Pythagorean Theorem:
- According to the Pythagorean theorem, in any right triangle, the sum of the squares of the lengths of the legs is equal to the square of the length of the hypotenuse. For triangle [tex]\(XYZ\)[/tex], with legs of length [tex]\(a\)[/tex], this can be written as:
[tex]\[
a^2 + a^2 = c^2.
\][/tex]
3. Combine Like Terms:
- Combine the terms on the left side of the equation:
[tex]\[
2a^2 = c^2.
\][/tex]
4. Determine the Principal Square Root of Both Sides:
- To solve for [tex]\(c\)[/tex], take the square root of both sides of the equation:
[tex]\[
\sqrt{2a^2} = \sqrt{c^2}.
\][/tex]
- Simplify the expressions:
[tex]\[
\sqrt{2} \cdot \sqrt{a^2} = c.
\][/tex]
[tex]\[
\sqrt{2} \cdot a = c.
\][/tex]
5. Result:
- Thus, the length of the hypotenuse [tex]\(c\)[/tex] is [tex]\(\sqrt{2}\)[/tex] times the length of each leg [tex]\(a\)[/tex]:
[tex]\[
c = \sqrt{2} \cdot a.
\][/tex]
Therefore, in a [tex]\(45^{\circ}-45^{\circ}-90^{\circ}\)[/tex] triangle, the hypotenuse is [tex]\(\sqrt{2}\)[/tex] times the length of each leg. The final step to show this relationship is to determine the principal square root of both sides of the equation [tex]\(2a^2 = c^2\)[/tex].
