Question

the triangle below is isosceles find the length of side x in simplest radical form with a rational denominator
the side beside the right angle is x the side across from it is 4

Answer 1

To solve for the length \( x \) in an isosceles triangle with a right angle, let’s establish what we know.

In an isosceles right triangle, the two legs are equal in length. Since you mentioned that the side beside the right angle is \( x \) and the side across from it (the hypotenuse) is \( 4 \), we can denote this as follows:

- Leg 1: \( x \)
- Leg 2: \( x \)
- Hypotenuse: \( 4 \)

Using the Pythagorean theorem, the relationship between the legs and the hypotenuse is represented as:

\[
x^2 + x^2 = 4^2
\]

This simplifies to:

\[
2x^2 = 16
\]

Now, divide both sides by 2:

\[
x^2 = 8
\]

Taking the square root of both sides gives:

\[
x = \sqrt{8} = \sqrt{4 \times 2} = 2\sqrt{2}
\]

Thus, the length of side \( x \) is

\[
\boxed{2\sqrt{2}}.
\]