To solve for $x$ in this right triangle, we can use the Pythagorean theorem. The Pythagorean theorem states:

$a^2 + b^2 = c^2$

where $c$ is the hypotenuse (the side opposite the right angle), and $a$ and $b$ are the other two sides of the triangle.

In this triangle:

• The hypotenuse $c = 65$
• One of the legs $b = 25$
• The other leg $a = x$ (which we need to find)

Applying the theorem:

$x^2 + 25^2 = 65^2$

1. First, calculate $25^2$ and $65^2$: $25^2 = 625$ $65^2 = 4225$

2. Substitute these values into the equation: $x^2 + 625 = 4225$

3. Subtract 625 from both sides: $x^2 = 3600$

4. Take the square root of both sides: $x = \sqrt{3600}$ $x = 60$

So, the length of $x$ is 60.

Would you like further details or have any questions?

Here are some related questions to explore further:

1. How would we solve for the other side if the hypotenuse and one leg were given?
2. What would change in the solution if the triangle had a different angle besides 90 degrees?
3. How does the Pythagorean theorem apply in three-dimensional space?
4. Can we solve this problem using trigonometric functions?
5. What is the significance of the Pythagorean triples in right triangles?

Tip: Always double-check calculations by substituting values back into the original equation.