Question

what is the length of y in this 45-45-90 triangle? (1 point) 4√2 8 8√2 4

Explanation:

Step1: Recall 45-45-90 triangle ratios

In a 45-45-90 triangle, the legs are equal, and the hypotenuse \( h \) is related to a leg \( l \) by \( h = l\sqrt{2} \), or \( l=\frac{h}{\sqrt{2}} \).

Step2: Identify hypotenuse and solve for leg \( y \)

The hypotenuse here is \( 8\sqrt{2} \). Let the leg length be \( y \). Using \( l=\frac{h}{\sqrt{2}} \), substitute \( h = 8\sqrt{2} \):
\[
y=\frac{8\sqrt{2}}{\sqrt{2}}
\]
Simplify: The \( \sqrt{2} \) terms cancel, so \( y = 8 \)? Wait, no, wait—wait, maybe I misread. Wait, no, in the triangle, the angle is 45°, so the two legs are equal (x and y), and hypotenuse is \( 8\sqrt{2} \). Wait, formula for 45-45-90: leg \( = \frac{\text{hypotenuse}}{\sqrt{2}} \). So:
\[
y=\frac{8\sqrt{2}}{\sqrt{2}} = 8? \text{ No, wait, no—wait, maybe the hypotenuse is } 8\sqrt{2}, \text{ so leg is } \frac{8\sqrt{2}}{\sqrt{2}} = 8? \text{ But the options have 8. Wait, maybe I made a mistake earlier. Wait, let's re-express:}
\]
Wait, \( \frac{8\sqrt{2}}{\sqrt{2}} = 8 \), because \( \sqrt{2}/\sqrt{2}=1 \), so \( 8 \times 1 = 8 \). Wait, but the initial selected option was \( 4\sqrt{2} \), which was wrong. Let's correct:

Wait, 45-45-90 triangle: legs are \( l \), hypotenuse \( l\sqrt{2} \). So if hypotenuse is \( 8\sqrt{2} \), then \( l\sqrt{2}=8\sqrt{2} \implies l = 8 \). So the leg \( y \) is 8.

Wait, maybe the diagram: the right angle, one angle 45°, so the two legs (x and y) are equal, hypotenuse is \( 8\sqrt{2} \). So solving for leg: \( l = \frac{\text{hypotenuse}}{\sqrt{2}} = \frac{8\sqrt{2}}{\sqrt{2}} = 8 \). So the correct answer is 8.

Answer:

B. 8 (assuming the option is "8" as one of the choices, likely the second option: "8")