Question

find c.
right triangle with angles 45°, 45°, 90°, one leg is \\(\sqrt{2}\\) in, hypotenuse is c
write your answer in simplest radical form.
blank inches

Explanation:

Step1: Identify triangle type

This is a 45 - 45 - 90 triangle (two 45° angles, one right angle), so it's isosceles right triangle. In a 45 - 45 - 90 triangle, the legs are equal, and the hypotenuse \( c \) is leg \( \times\sqrt{2} \), or leg \( = \frac{c}{\sqrt{2}} \). Here, one leg is \( \sqrt{2} \) in.

Step2: Use Pythagorean theorem (or 45 - 45 - 90 ratio)

For a 45 - 45 - 90 triangle, if leg length is \( a \), hypotenuse \( c=a\sqrt{2} \). Wait, or since legs are equal, both legs are \( \sqrt{2} \) in. Then by Pythagorean theorem: \( c^{2}=(\sqrt{2})^{2}+(\sqrt{2})^{2} \)
Calculate \( (\sqrt{2})^{2}=2 \), so \( c^{2}=2 + 2=4 \), then \( c = \sqrt{4}=2 \). Or using the ratio: in 45 - 45 - 90, hypotenuse is leg \( \times\sqrt{2} \), but wait, no—wait, if leg is \( \sqrt{2} \), then hypotenuse \( c=\) leg \( \times\sqrt{2} \)? Wait no, wait the ratio is leg : leg : hypotenuse \( = 1:1:\sqrt{2} \). So if leg is \( x \), hypotenuse is \( x\sqrt{2} \). But here leg is \( \sqrt{2} \), so hypotenuse \( c=\sqrt{2}\times\sqrt{2}=2 \). Yes, that's correct.

Answer:

\( 2 \)