Consider the incomplete paragraph proof.

Given: Isosceles right triangle XYZ [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.



Answer :

To prove that in a [tex]\(45^{\circ}-45^{\circ}-90^{\circ}\)[/tex] triangle the hypotenuse [tex]\(c\)[/tex] is [tex]\(\sqrt{2}\)[/tex] times the length of each leg [tex]\(a\)[/tex], follow these steps:

1. Understand the Triangle:
- In a [tex]\(45^{\circ}-45^{\circ}-90^{\circ}\)[/tex] triangle, two legs are of equal length, let's call this length [tex]\(a\)[/tex]. The triangle is isosceles, meaning the legs have the same length, and the angles opposite these legs are each [tex]\(45^{\circ}\)[/tex].

2. Apply the Pythagorean Theorem:
- According to the Pythagorean theorem, in any right triangle, the sum of the squares of the lengths of the legs is equal to the square of the length of the hypotenuse. For triangle [tex]\(XYZ\)[/tex], with legs of length [tex]\(a\)[/tex], this can be written as:
[tex]\[ a^2 + a^2 = c^2. \][/tex]

3. Combine Like Terms:
- Combine the terms on the left side of the equation:
[tex]\[ 2a^2 = c^2. \][/tex]

4. Determine the Principal Square Root of Both Sides:
- To solve for [tex]\(c\)[/tex], take the square root of both sides of the equation:
[tex]\[ \sqrt{2a^2} = \sqrt{c^2}. \][/tex]
- Simplify the expressions:
[tex]\[ \sqrt{2} \cdot \sqrt{a^2} = c. \][/tex]
[tex]\[ \sqrt{2} \cdot a = c. \][/tex]

5. Result:
- Thus, the length of the hypotenuse [tex]\(c\)[/tex] is [tex]\(\sqrt{2}\)[/tex] times the length of each leg [tex]\(a\)[/tex]:
[tex]\[ c = \sqrt{2} \cdot a. \][/tex]

Therefore, in a [tex]\(45^{\circ}-45^{\circ}-90^{\circ}\)[/tex] triangle, the hypotenuse is [tex]\(\sqrt{2}\)[/tex] times the length of each leg. The final step to show this relationship is to determine the principal square root of both sides of the equation [tex]\(2a^2 = c^2\)[/tex].