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Consider the incomplete paragraph proof.

Given: Isosceles right triangle [tex]$XYZ$[/tex] (45°-45°-90° triangle)

Prove: In a 45°-45°-90° 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 :

Let's solve the problem step-by-step:

1. Given: An isosceles right triangle \(XYZ\) with angles \(45^\circ\), \(45^\circ\), and \(90^\circ\).

2. Objective: To prove that in a \(45^\circ\)-\(45^\circ\)-\(90^\circ\) (isosceles right) triangle, the hypotenuse is \(\sqrt{2}\) times the length of each leg.

3. Pythagorean Theorem Application: In any right triangle, the side lengths must satisfy the Pythagorean Theorem: \( a^2 + b^2 = c^2 \).

Since triangle \(XYZ\) is isosceles:
- Let \(a\) be the length of each leg (since both legs are of equal length).
- Let \(c\) be the length of the hypotenuse.

Given the isosceles property, we substitute \(a\) for both legs:
[tex]\[ a^2 + a^2 = c^2 \][/tex]

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

5. Determine the principal square root of both sides:
To isolate \(c\), we need to solve for \(c\) by taking the principal square root of both sides:
[tex]\[ c = \sqrt{2a^2} \][/tex]

6. Simplify the Square Root: Since \(\sqrt{2a^2} = \sqrt{2} \cdot \sqrt{a^2}\), we get:
[tex]\[ c = \sqrt{2} \cdot a \][/tex]

Thus, we have proved that the length of the hypotenuse \(c\) is \(\sqrt{2}\) times the length of each leg \(a\).

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