Question

which is a true statement about a 45 - 45 - 90 triangle? a. the hypotenuse is √3 times as long as either leg. b. the hypotenuse is √2 times as long as either leg. c. each leg is √2 times as long as the hypotenuse. d. each leg is √3 times as long as the hypotenuse.

Explanation:

Step1: Recall Pythagorean theorem

For a right - triangle with legs of length $a$ and $b$ and hypotenuse $c$, $a^{2}+b^{2}=c^{2}$. In a 45 - 45 - 90 triangle, $a = b$.

Step2: Substitute $a = b$ into Pythagorean theorem

If $a = b$, then $a^{2}+a^{2}=c^{2}$, which simplifies to $2a^{2}=c^{2}$.

Step3: Solve for $c$ in terms of $a$

Taking the square root of both sides of $2a^{2}=c^{2}$, we get $c=\sqrt{2a^{2}}=\sqrt{2}a$. This means the hypotenuse $c$ is $\sqrt{2}$ times as long as either leg $a$.

Answer:

B. The hypotenuse is $\sqrt{2}$ times as long as either leg.