To determine which statement is true about an isosceles right triangle, let's analyze the properties of this specific type of triangle.
An isosceles right triangle has two legs of equal length and one hypotenuse. If the length of each leg is denoted by [tex]\(a\)[/tex], the Pythagorean theorem helps us to find the length of the hypotenuse ([tex]\(c\)[/tex]).
The Pythagorean theorem states that in a right triangle:
[tex]\[
a^2 + b^2 = c^2
\][/tex]
Since both legs ([tex]\(a\)[/tex] and [tex]\(b\)[/tex]) are equal in an isosceles right triangle, we can write:
[tex]\[
a^2 + a^2 = c^2 \implies 2a^2 = c^2
\][/tex]
Solving for [tex]\(c\)[/tex]:
[tex]\[
c^2 = 2a^2 \implies c = \sqrt{2a^2} \implies c = a\sqrt{2}
\][/tex]
Therefore, the hypotenuse [tex]\(c\)[/tex] is [tex]\(\sqrt{2}\)[/tex] times the length of either leg.
Let's evaluate each option:
A. The hypotenuse is [tex]\(\sqrt{2}\)[/tex] times as long as either leg.
[tex]\[
\text{This statement is true based on our derived formula } c = a\sqrt{2}.
\][/tex]
B. Each leg is [tex]\(\sqrt{3}\)[/tex] times as long as the hypotenuse.
[tex]\[
\text{This statement is false because based on our derived formula we have } a = \frac{c}{\sqrt{2}}, \text{ not } \frac{c}{\sqrt{3}}.
\][/tex]
C. The hypotenuse is [tex]\(\sqrt{3}\)[/tex] times as long as either leg.
[tex]\[
\text{This statement is false because the hypotenuse is actually } \sqrt{2} \text{ times as long as either leg.}
\][/tex]
D. Each leg is [tex]\(\sqrt{2}\)[/tex] times as long as the hypotenuse.
[tex]\[
\text{This statement is false because we derived } c = a\sqrt{2}, \text{ so each leg is } \frac{c}{\sqrt{2}}, \text{ not } \sqrt{2} \text{ times the hypotenuse.}
\][/tex]
Based on our analysis, the correct statement is:
[tex]\[
\boxed{A}
\][/tex]
