Welcome to Westonci.ca, the place where your questions find answers from a community of knowledgeable experts. Ask your questions and receive detailed answers from professionals with extensive experience in various fields. Get quick and reliable solutions to your questions from a community of experienced experts on our platform.

Consider the incomplete paragraph proof.

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

Sagot :

To complete the 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, we can follow the steps as outlined below:

Given:
- Triangle [tex]\(XYZ\)[/tex] is an isosceles right triangle, also known as a [tex]\(45^{\circ}-45^{\circ}-90^{\circ}\)[/tex] triangle.
- Let the legs of the triangle be [tex]\(a\)[/tex] and [tex]\(a\)[/tex], and the hypotenuse be [tex]\(c\)[/tex].

Step-by-Step Solution:

1. Initial Equation:
We start by using the Pythagorean theorem for the right triangle [tex]\(XYZ\)[/tex]:
[tex]\[ a^2 + b^2 = c^2 \][/tex]
Since [tex]\(XYZ\)[/tex] is an isosceles right triangle, the lengths of the legs are equal, so [tex]\(a = b\)[/tex]. Thus, the equation becomes:
[tex]\[ a^2 + a^2 = c^2 \][/tex]

2. Combine Like Terms:
By combining the terms on the left side of the equation, we get:
[tex]\[ 2a^2 = c^2 \][/tex]

3. Taking the Square Root:
To find the length of the hypotenuse [tex]\(c\)[/tex], we need to determine the principal square root of both sides of the equation:
[tex]\[ c = \sqrt{2a^2} \][/tex]

4. Simplify the Square Root:
Simplifying the square root expression, we can separate the constant factor and the variable:
[tex]\[ c = \sqrt{2} \cdot \sqrt{a^2} \][/tex]
Since [tex]\(\sqrt{a^2} = a\)[/tex], the equation becomes:
[tex]\[ c = \sqrt{2} \cdot a \][/tex]

Therefore, we have proven that in a [tex]\(45^{\circ}-45^{\circ}-90^{\circ}\)[/tex] triangle, the length of the hypotenuse [tex]\(c\)[/tex] is [tex]\(\sqrt{2}\)[/tex] times the length of each leg [tex]\(a\)[/tex]. The final answer is:
[tex]\[ c = \sqrt{2} \cdot a \][/tex]

This completes the proof with the final result:
[tex]\[ c = \sqrt{2}a \][/tex]