To determine the true statement about an isosceles right triangle, let's investigate the properties of an isosceles right triangle step-by-step.
An isosceles right triangle has two sides (legs) of the same length and one side (the hypotenuse) that is longer. The right angle is between the two legs.
1. Consider a typical isosceles right triangle where each leg has a length of [tex]\( a \)[/tex].
2. Using the Pythagorean theorem:
[tex]\[
\text{Hypotenuse}^2 = \text{Leg}^2 + \text{Leg}^2
\][/tex]
[tex]\[
\text{Hypotenuse}^2 = a^2 + a^2
\][/tex]
[tex]\[
\text{Hypotenuse}^2 = 2a^2
\][/tex]
3. Taking the square root of both sides to solve for the hypotenuse:
[tex]\[
\text{Hypotenuse} = \sqrt{2a^2}
\][/tex]
[tex]\[
\text{Hypotenuse} = a\sqrt{2}
\][/tex]
4. Therefore, the hypotenuse is:
[tex]\[
\sqrt{2} \text{ times as long as either leg.}
\][/tex]
Given the analysis, we can see that the correct answer is:
D. The hypotenuse is [tex]\(\sqrt{2}\)[/tex] times as long as either leg.
