To solve this problem, let's analyze the information given and apply the Pythagorean theorem.

Since the triangle is isosceles and has a right angle, it is a right isosceles triangle. This type of triangle has two equal legs and one hypotenuse. Given:

• One leg has length $x$,
• The hypotenuse has length $10$.

Solution:

In a right isosceles triangle, the legs are equal. By the Pythagorean theorem: $x^2 + x^2 = 10^2$

Simplify: $2x^2 = 100$

Divide by 2: $x^2 = 50$

Take the square root of both sides: $x = \sqrt{50} = \sqrt{25 \times 2} = 5\sqrt{2}$

Final Answer:

The length of side $x$ is $5\sqrt{2}$.

Would you like further details on this solution, or do you have any questions?

Here are five related questions for further practice:

1. What would the length of $x$ be if the hypotenuse were 20 instead?
2. How do you find the hypotenuse if both legs are given?
3. What properties do isosceles right triangles have?
4. How would you solve for $x$ if the triangle weren’t isosceles?
5. Can this method be applied to any right triangle?

Tip: In an isosceles right triangle, the hypotenuse is always $\sqrt{2}$ times the length of each leg.