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Consider the incomplete paragraph proof.

Given: Isosceles right triangle \(XYZ \left(45^{\circ}-45^{\circ}-90^{\circ}\right.\) triangle)

Prove: In a \(45^{\circ}-45^{\circ}-90^{\circ}\) triangle, the hypotenuse is \(\sqrt{2}\) times the length of each leg.

Because triangle \(XYZ\) is a right triangle, the side lengths must satisfy the Pythagorean theorem, \(a^2 + b^2 = c^2\), which in this isosceles triangle becomes \(a^2 + a^2 = c^2\). By combining like terms, \(2a^2 = c^2\).

Which final step will prove that the length of the hypotenuse, \(c\), is \(\sqrt{2}\) times the length of each leg?

A. Substitute values for \(a\) and \(c\) 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
Certainly! Let's complete the proof step-by-step:

Given:
Isosceles right triangle \( XYZ \) is a \( 45^{\circ}-45^{\circ}-90^{\circ} \) triangle.

To Prove:
In a \( 45^{\circ}-45^{\circ}-90^{\circ} \) triangle, the hypotenuse (\( c \)) is \( \sqrt{2} \) times the length of each leg (\( a \)).

1. State the Pythagorean Theorem:
Since triangle \( XYZ \) is a right triangle, the side lengths must satisfy the Pythagorean theorem:
[tex]\[ a^2 + b^2 = c^2 \][/tex]

2. Account for Isosceles Triangle Properties:
In an isosceles right triangle, the legs \( a \) and \( b \) are of equal length:
[tex]\[ a = b \][/tex]

3. Substitute \( b \) with \( a \) in the Pythagorean Theorem:
[tex]\[ a^2 + a^2 = c^2 \][/tex]

4. Combine Like Terms:
[tex]\[ 2a^2 = c^2 \][/tex]

5. Divide Both Sides by 2:
To isolate \( a^2 \):
[tex]\[ a^2 = \frac{c^2}{2} \][/tex]

6. Solve for \( c \):
To express \( c \) in terms of \( a \), take the principal square root of both sides:
[tex]\[ c = \sqrt{2a^2} \][/tex]

7. Simplify the Expression:
[tex]\[ c = \sqrt{2} \cdot a \][/tex]

Conclusion:
The length of 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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