To finalize the proof 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], we need to perform and explain the following steps:
1. Set Up the Pythagorean Theorem:
For the isosceles right triangle [tex]\(XYZ\)[/tex], which has two legs of equal length [tex]\(a\)[/tex], the Pythagorean theorem states:
[tex]\[
a^2 + a^2 = c^2
\][/tex]
Since both legs are of equal length, substitute [tex]\(a\)[/tex] for both [tex]\(a\)[/tex] and [tex]\(b\)[/tex]:
2. Combine Like Terms:
Combine the like terms on the left side of the equation:
[tex]\[
a^2 + a^2 = 2a^2
\][/tex]
This simplifies the equation to:
[tex]\[
2a^2 = c^2
\][/tex]
3. Find the Principal Square Root:
To solve for [tex]\(c\)[/tex] in terms of [tex]\(a\)[/tex], we need to determine the principal square root of both sides of the equation:
[tex]\[
\sqrt{2a^2} = \sqrt{c^2}
\][/tex]
Simplifying both sides, we get:
[tex]\[
\sqrt{2} \cdot a = c
\][/tex]
Therefore, this final step proves that the hypotenuse [tex]\(c\)[/tex] is indeed [tex]\(\sqrt{2}\)[/tex] times the length of each leg [tex]\(a\)[/tex]:
[tex]\[
c = \sqrt{2} \cdot a
\][/tex]
The correct final step is to determine the principal square root of both sides of the equation [tex]\(2a^2 = c^2\)[/tex].
