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Because triangle XYZ is a right triangle, the side lengths must satisfy the Pythagorean theorem, [tex]a^2 + b^2 = c^2[/tex]. In this isosceles triangle, it 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 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 the length of the hypotenuse [tex]\( c \)[/tex] is [tex]\(\sqrt{2}\)[/tex] times the length of each leg [tex]\( a \)[/tex] in the given isosceles right triangle, follow these steps:

1. Start with the equation derived from the Pythagorean theorem for the given isosceles right triangle:
[tex]\[ a^2 + a^2 = c^2 \][/tex]

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

3. To isolate [tex]\( a^2 \)[/tex], divide both sides of the equation by 2:
[tex]\[ a^2 = \frac{c^2}{2} \][/tex]

4. Now, to solve for [tex]\( a \)[/tex], determine the principal square root of both sides of the equation:
[tex]\[ a = \sqrt{\frac{c^2}{2}} \][/tex]

5. Since [tex]\(\sqrt{\frac{c^2}{2}} = \frac{c}{\sqrt{2}}\)[/tex], this step demonstrates that:
[tex]\[ a = \frac{c}{\sqrt{2}} \][/tex]

6. By multiplying both sides by [tex]\(\sqrt{2}\)[/tex], we get:
[tex]\[ a \sqrt{2} = c \][/tex]

Therefore, the final step that confirms that the length of the hypotenuse [tex]\( c \)[/tex] is indeed [tex]\(\sqrt{2}\)[/tex] times the length of each leg [tex]\( a \)[/tex] is:

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