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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 XYZ 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.

Sagot :

GinnyAnswer
To prove 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, follow these steps:

1. Start with the Pythagorean theorem for the right triangle:
In any right triangle, the side lengths satisfy the Pythagorean theorem: [tex]\(a^2 + b^2 = c^2\)[/tex].

2. Apply the Pythagorean theorem to our specific isosceles right triangle:
Since it is a [tex]\(45^{\circ}-45^{\circ}-90^{\circ}\)[/tex] triangle, the legs are equal in length. Let's denote each leg as [tex]\(a\)[/tex]. Thus, the equation becomes [tex]\(a^2 + a^2 = c^2\)[/tex].

3. Combine like terms:
This simplifies to:
[tex]\[ 2a^2 = c^2 \][/tex]

4. Isolate [tex]\(a^2\)[/tex] by dividing both sides by 2:
[tex]\[ a^2 = \frac{c^2}{2} \][/tex]

5. Determine the principal square root of both sides:
Take the square root of both sides of the equation to solve for [tex]\(a\)[/tex]:
[tex]\[ a = \sqrt{\frac{c^2}{2}} \][/tex]

6. Simplify the square root expression:
[tex]\[ a = \frac{c}{\sqrt{2}} \][/tex]

7. Solve for [tex]\(c\)[/tex] in terms of [tex]\(a\)[/tex]:
Multiply both sides by [tex]\(\sqrt{2}\)[/tex]:
[tex]\[ c = a \sqrt{2} \][/tex]

Therefore, the hypotenuse [tex]\(c\)[/tex] is [tex]\(\sqrt{2}\)[/tex] times the length of each leg [tex]\(a\)[/tex] in a [tex]\(45^{\circ}-45^{\circ}-90^{\circ}\)[/tex] triangle. This completes the proof.
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