Let's complete the proof step-by-step to show that the hypotenuse [tex]\(c\)[/tex] in a [tex]\(45^\circ-45^\circ-90^\circ\)[/tex] triangle is [tex]\(\sqrt{2}\)[/tex] times the length of each leg [tex]\(a\)[/tex].
1. Restate the Given Information:
We have an isosceles right triangle [tex]\(XYZ\)[/tex] with angles [tex]\(45^\circ, 45^\circ\)[/tex], and [tex]\(90^\circ\)[/tex].
2. Apply the Pythagorean Theorem:
For right triangle [tex]\(XYZ\)[/tex]:
[tex]\[
a^2 + b^2 = c^2
\][/tex]
Since [tex]\(XYZ\)[/tex] is isosceles, the legs [tex]\(a\)[/tex] and [tex]\(b\)[/tex] are equal:
[tex]\[
a^2 + a^2 = c^2
\][/tex]
3. Combine Like Terms:
Simplify the equation:
[tex]\[
2a^2 = c^2
\][/tex]
4. Determine the Principal Square Root of Both Sides:
To isolate [tex]\(c\)[/tex], take the square root of both sides:
[tex]\[
\sqrt{2a^2} = \sqrt{c^2}
\][/tex]
Simplify the square roots:
[tex]\[
a\sqrt{2} = c
\][/tex]
Therefore, the length of the hypotenuse [tex]\(c\)[/tex] is [tex]\(\sqrt{2}\)[/tex] times the length of each leg [tex]\(a\)[/tex]:
[tex]\[
c = a\sqrt{2}
\][/tex]
This completes our 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.
