https://file.aiquickdraw.com/mathbot/file/1730703711448dkgtfmqv.jpg

To solve this problem, let's analyze the isosceles triangle shown.

### Problem Breakdown
1. **Triangle Properties**: The triangle is isosceles, meaning two of its sides are equal in length. 
2. **Right Angle**: There is a right angle marked in the triangle, so it’s also a right triangle.
3. **Sides Given**: The leg on one side of the right angle is labeled with a length of \( \sqrt{10} \), and the hypotenuse is labeled as \( x \).

### Solution
Since this is a right triangle, we can apply the **Pythagorean theorem**, which states:
\[
a^2 + b^2 = c^2
\]
where \( c \) is the hypotenuse, and \( a \) and \( b \) are the legs of the triangle.

In an isosceles right triangle, the two legs are equal. This means both legs of the triangle are \( \sqrt{10} \).

Let’s apply the Pythagorean theorem:
\[
(\sqrt{10})^2 + (\sqrt{10})^2 = x^2
\]
Calculating each term:
1. \( (\sqrt{10})^2 = 10 \)
2. So,
   \[
   10 + 10 = x^2
   \]
   \[
   x^2 = 20
   \]
3. Taking the square root of both sides to solve for \( x \):
   \[
   x = \sqrt{20} = \sqrt{4 \cdot 5} = 2\sqrt{5}
   \]

### Final Answer
The length of side \( x \) in simplest radical form is:
\[
x = 2\sqrt{5}
\]

Would you like additional explanation on any part of this solution?

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Here are five related questions you may find helpful:
1. How would you solve for the hypotenuse if the legs were different lengths?
2. What other properties do isosceles triangles have?
3. How does the Pythagorean theorem apply to non-isosceles triangles?
4. What is the process of simplifying square roots in radical form?
5. How does identifying a triangle as isosceles help solve problems?

**Tip**: Remember, in a 45°-45°-90° triangle (which is an isosceles right triangle), the hypotenuse is always \( \sqrt{2} \) times the length of each leg.

The triangle below is isosceles. Find the length of side x in simplest radical form with a rational denominator.

Learn how to find the length of side x in an isosceles right triangle using the Pythagorean theorem and radical simplification. This problem covers key geometric concepts and is suitable for students in Grades 8-10.

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