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To find the length of the third side of the right triangle, we can use the Pythagorean theorem:

\[
a^2 + b^2 = c^2
\]

where:
- \( a \) and \( b \) are the lengths of the two legs,
- \( c \) is the length of the hypotenuse (the side opposite the right angle).

From the diagram:
- One leg is 9,
- The other leg (hypotenuse) is 13.

Since we have the hypotenuse, we’ll use the formula as follows:

\[
9^2 + b^2 = 13^2
\]

Calculating each term:

1. \( 9^2 = 81 \)
2. \( 13^2 = 169 \)

Substitute these into the equation:

\[
81 + b^2 = 169
\]

Now, solve for \( b^2 \):

\[
b^2 = 169 - 81
\]
\[
b^2 = 88
\]

Now, take the square root of both sides:

\[
b = \sqrt{88} \approx 9.4
\]

So, the length of the third side is approximately **9.4**, rounded to the nearest tenth.

Would you like further details or have any questions?

Here are five related questions you may find helpful:
1. How do you apply the Pythagorean theorem to different types of triangles?
2. How do you determine if a triangle is a right triangle using side lengths?
3. What is the relationship between the sides of a 45-45-90 triangle?
4. How would you solve for a missing side if only the hypotenuse and one leg are known?
5. What are some real-world applications of the Pythagorean theorem?

**Tip:** Always double-check that the hypotenuse is the longest side when using the Pythagorean theorem in right triangles.

Find the length of the third side of the right triangle with sides of lengths 9 and 13.

Calculate the length of the third side in a right triangle with given side lengths of 9 and 13. Learn to apply the Pythagorean Theorem for solving right triangles, a core geometry concept for students in Grades 8-10.

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