To solve for the length of side $x$ in this isosceles right triangle, we can apply the Pythagorean theorem or use properties specific to an isosceles right triangle.

Step 1: Analyze the Triangle

Since the triangle is isosceles and right-angled, the two legs of the triangle are equal in length. The given leg is 10 units long, and the hypotenuse is $x$.

In an isosceles right triangle, the legs are related to the hypotenuse using the ratio $1:1:\sqrt{2}$. This means: $\text{Hypotenuse} = \text{Leg} \times \sqrt{2}$ Therefore: $x = 10 \times \sqrt{2}$

Step 2: Simplify the Expression

The length of side $x$ is: $x = 10\sqrt{2}$

Thus, the length of side $x$ in simplest radical form with a rational denominator is $10\sqrt{2}$.

Would you like further details on the solution? Feel free to ask any questions. Here are some follow-up topics you might find useful:

1. How does the Pythagorean theorem apply to right-angled triangles?
2. How do the properties of isosceles triangles help simplify problems?
3. The relationship between the legs and hypotenuse in special right triangles.
4. How to rationalize a denominator in radical expressions.
5. Solving for unknown sides in other types of triangles.

Tip: Remember that in an isosceles right triangle, the ratio of the sides is always $1:1:\sqrt{2}$. This helps in quickly finding the length of the hypotenuse given the legs!