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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
The given information tells us that [tex]$XYZ$[/tex] is an isosceles right triangle, specifically a [tex]$45^{\circ}-45^{\circ}-90^{\circ}$[/tex] triangle. To prove that the hypotenuse ([tex]$c$[/tex]) is [tex]$\sqrt{2}$[/tex] times the length of each leg ([tex]$a$[/tex]), we start by recognizing that the triangle adheres to the Pythagorean theorem.

1. Establishing the Pythagorean theorem for this triangle:
Since [tex]$XYZ$[/tex] is a 45°-45°-90° triangle, both legs (let's call them [tex]$a$[/tex]) are equal:
[tex]\[ a^2 + a^2 = c^2 \][/tex]

2. Combining like terms:
Adding [tex]$a^2 + a^2$[/tex] gives us [tex]$2a^2$[/tex], so we have:
[tex]\[ 2a^2 = c^2 \][/tex]

3. Solving for [tex]$c$[/tex]:
To find [tex]$c$[/tex], we need to isolate [tex]$c$[/tex] on one side of the equation. First, we divide both sides by 2:
[tex]\[ a^2 = \frac{c^2}{2} \][/tex]

4. Taking the principal square root of both sides:
To find [tex]$c$[/tex], we take the square root of both sides:
[tex]\[ a = \frac{c}{\sqrt{2}} \][/tex]
Multiplying both sides by [tex]$\sqrt{2}$[/tex] to solve for [tex]$c$[/tex]:
[tex]\[ c = a\sqrt{2} \][/tex]

Thus, we have shown 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 either leg [tex]$a$[/tex]:
[tex]\[ c = a\sqrt{2} \][/tex]

This completes the proof.

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