https://file.aiquickdraw.com/mathbot/file/1728769240180eu03dhay.jpg

This is a right-angled triangle, and we can use the **Pythagorean theorem** to find the unknown side, \(x\).

The Pythagorean theorem states:
\[
a^2 + b^2 = c^2
\]
where \(a\) and \(b\) are the legs of the triangle, and \(c\) is the hypotenuse (the side opposite the right angle).

In this case:
- One leg, \(a = x\) (the unknown side).
- The other leg, \(b = 6 \, \text{cm}\).
- The hypotenuse, \(c = 13 \, \text{cm}\).

So, the equation becomes:
\[
x^2 + 6^2 = 13^2
\]
\[
x^2 + 36 = 169
\]
\[
x^2 = 169 - 36
\]
\[
x^2 = 133
\]
\[
x = \sqrt{133} \approx 11.54 \, \text{cm}
\]

Thus, the length of \(x\) is approximately **11.54 cm**.

Would you like more details on any part of this solution? 

Here are 5 related questions you could explore:
1. What is the Pythagorean theorem used for in real-world applications?
2. How do you determine the hypotenuse in any right triangle?
3. How can we find the length of the other leg if only one leg and the hypotenuse are known?
4. What are some ways to simplify square roots like \( \sqrt{133} \)?
5. How would this calculation change if the triangle wasn’t a right triangle?

**Tip:** The Pythagorean theorem is only applicable in right-angled triangles! For other types of triangles, you would need different formulas.

Work out the length x in the right-angled triangle with sides 13 cm and 6 cm.

Solve for the unknown side (x) of a right-angled triangle with sides 13 cm and 6 cm using the Pythagorean Theorem. This solution involves applying the formula a^2 + b^2 = c^2 and finding that the length of x is approximately 11.54 cm. Suitable for Grades 8-10 geometry students.

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