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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?

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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.

Find the missing length indicated (x) in the right triangle with sides of length 25 and 65.

Learn how to find the missing length x in a right triangle with sides of 25 and 65. This solution applies the Pythagorean Theorem, guiding you through the step-by-step process to determine x = 60. Suitable for Grades 8-10.

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