Answered

Proof:

For a 45°-45°-90° triangle, the hypotenuse is √2 times the length of each leg.

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

1. Start with the Pythagorean Theorem:
For a right triangle, the Pythagorean theorem states that [tex]\( a^2 + b^2 = c^2 \)[/tex], where [tex]\( a \)[/tex] and [tex]\( b \)[/tex] are the legs of the triangle and [tex]\( c \)[/tex] is the hypotenuse.

2. Isosceles Right Triangle Properties:
In a [tex]\( 45^\circ-45^\circ-90^\circ \)[/tex] triangle, the legs are of equal length. Let's denote the length of each leg by [tex]\( a \)[/tex]. Hence, the equation becomes:
[tex]\[ a^2 + a^2 = c^2 \][/tex]

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

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

5. Take the Principal Square Root:
Determine the principal square root of both sides of the equation:
[tex]\[ a = \frac{c}{\sqrt{2}} \][/tex]

Since we want to express [tex]\( c \)[/tex] in terms of [tex]\( a \)[/tex]:
[tex]\[ c = a \times \sqrt{2} \][/tex]

6. Conclusion:
We have shown that the length of the hypotenuse [tex]\( c \)[/tex] is [tex]\( \sqrt{2} \)[/tex] times the length of each leg [tex]\( a \)[/tex].

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