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Consider the incomplete paragraph proof.

Given: Isosceles right triangle [tex]$XYZ \left(45^{\circ}-45^{\circ}-90^{\circ}\right.$[/tex] triangle)

Prove: In a [tex]$45^{\circ}-45^{\circ}-90^{\circ}$[/tex] triangle, the hypotenuse is [tex]$\sqrt{2}$[/tex] times the length of each leg.

Because triangle [tex]$XYZ$[/tex] is a right triangle, the side lengths must satisfy the Pythagorean theorem, [tex]$a^2 + b^2 = c^2$[/tex], which in this isosceles triangle becomes [tex]$a^2 + a^2 = c^2$[/tex]. By combining like terms, [tex]$2a^2 = c^2$[/tex].

Which final step will prove that the length of the hypotenuse, [tex]$c$[/tex], is [tex]$\sqrt{2}$[/tex] times the length of each leg?

A. Substitute values for [tex]$a$[/tex] and [tex]$c$[/tex] into the original Pythagorean theorem equation.
B. Divide both sides of the equation by two, then determine the principal square root of both sides of the equation.
C. Determine the principal square root of both sides of the equation.
D. Divide both sides of the equation by 2.



Answer :

GinnyAnswer
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].

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