https://file.aiquickdraw.com/mathbot/file/1729020045123c1od9x5s.jpg

The image shows a right triangle with side lengths, one being \( 11 \) and the hypotenuse labeled as \( x \). This triangle is an isosceles right triangle, meaning that the two legs are congruent. Since this is a right triangle, we can apply the Pythagorean Theorem to find the value of \( x \) in simplest radical form.

The Pythagorean Theorem states:

\[
a^2 + b^2 = c^2
\]

where \( a \) and \( b \) are the legs of the triangle, and \( c \) is the hypotenuse. Here, one leg is \( 11 \), and the other leg is also \( 11 \) because the triangle is isosceles. The hypotenuse is \( x \).

Substituting the known values into the Pythagorean Theorem:

\[
11^2 + 11^2 = x^2
\]

\[
121 + 121 = x^2
\]

\[
242 = x^2
\]

To find \( x \), take the square root of both sides:

\[
x = \sqrt{242}
\]

Simplifying \( \sqrt{242} \):

\[
x = \sqrt{121 \times 2} = \sqrt{121} \times \sqrt{2} = 11\sqrt{2}
\]

Thus, the length of side \( x \) in simplest radical form is:

\[
x = 11\sqrt{2}
\]

Would you like more details, or do you have any questions?

### 5 Related Questions:
1. How is the Pythagorean Theorem applied in non-right isosceles triangles?
2. What are other methods to simplify radicals?
3. Can this problem be solved using trigonometry instead of the Pythagorean Theorem?
4. How would the problem change if the triangle was not isosceles?
5. What real-world applications involve using the Pythagorean Theorem?

### Tip:
Always check if a radical can be simplified by factoring the number under the square root into its prime factors.

Find the length of side x in simplest radical form with a rational denominator. The triangle has a leg of length 11 and the hypotenuse is labeled x.

Solve for side length x in a right triangle where one leg is 11 units, using the Pythagorean Theorem and expressing the result in simplest radical form. This geometry problem involves simplifying radicals and applying basic theorems.

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