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Consider the incomplete paragraph proof.

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

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 2, 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 :

GinnyAnswer
To prove that in a \( 45^\circ-45^\circ-90^\circ \) triangle the hypotenuse is \(\sqrt{2}\) times the length of each leg, we are given that the triangle is an isosceles right triangle where the legs have equal length \(a\). The proof should proceed by using the Pythagorean theorem and then finding the relationship between the hypotenuse \(c\) and the legs \(a\).

Given:
[tex]\[ a^2 + a^2 = c^2 \][/tex]

Combine like terms on the left side:
[tex]\[ 2a^2 = c^2 \][/tex]

We need to isolate \(c\). For this, we take the principal (positive) square root of both sides of the equation:
[tex]\[ \sqrt{2a^2} = \sqrt{c^2} \][/tex]

Since \(\sqrt{c^2} = c\) and \(\sqrt{2a^2} = \sqrt{2} \cdot \sqrt{a^2} = \sqrt{2} \cdot a\):
[tex]\[ c = \sqrt{2} \cdot a \][/tex]

This final step shows that the length of the hypotenuse \(c\) is \(\sqrt{2}\) times the length of each leg \(a\). Hence, the correct step is to:

Determine the principal square root of both sides of the equation.