https://file.aiquickdraw.com/mathbot/file/1725933696573xa3l58d2.jpg

The image contains two right-angled triangles, and the task is to determine the length of side \( AB \) for each triangle.

### Triangle 1 (Top):
- We are given:
  - \( AC = 6 \, \text{cm} \) (one leg),
  - \( CB = 15 \, \text{cm} \) (hypotenuse),
  - \( CA \perp AB \) (indicating a right-angled triangle at \( A \)).

We can use the **Pythagorean theorem** to find side \( AB \):
\[
AB^2 + AC^2 = CB^2
\]
Substituting the known values:
\[
AB^2 + 6^2 = 15^2
\]
\[
AB^2 + 36 = 225
\]
\[
AB^2 = 225 - 36 = 189
\]
\[
AB = \sqrt{189} \approx 13.75 \, \text{cm}
\]

So, \( AB \approx 13.75 \, \text{cm} \).

### Triangle 2 (Bottom):
- We are given:
  - \( CE = 12 \, \text{cm} \) (one leg),
  - \( BE = 20 \, \text{cm} \) (hypotenuse),
  - \( CA = 15 \, \text{cm} \) (one leg).

We can also use the **Pythagorean theorem** here for triangle \( BCE \):
\[
CE^2 + BE^2 = CB^2
\]
Substituting the known values:
\[
12^2 + 15^2 = 20^2
\]
\[
144 + 225 = 400
\]
This verifies that the values satisfy the Pythagorean theorem, meaning the triangle is valid.

So, in both triangles:
- Triangle 1: \( AB \approx 13.75 \, \text{cm} \),
- Triangle 2: Verified dimensions.

Would you like to go into further details, or have any specific questions?

### Related Questions:
1. What is the Pythagorean theorem, and how does it apply to right triangles?
2. How can you calculate the hypotenuse if both legs are known?
3. Can the Pythagorean theorem be applied to non-right triangles?
4. What are other methods to solve triangles apart from the Pythagorean theorem?
5. How can trigonometric ratios (like sine or cosine) be used in these triangles?

**Tip:** The Pythagorean theorem is fundamental for solving right triangles, but make sure to confirm the triangle's right angle before applying it.

Determine the length of side AB for each triangle in the diagram provided. The top triangle has sides AC = 6 cm, CB = 15 cm, and a right angle at A. The bottom triangle has sides CE = 12 cm, BE = 20 cm, and a right angle at E.

This problem involves finding the length of side AB in two right-angled triangles using the Pythagorean Theorem. In the first triangle, sides AC = 6 cm and CB = 15 cm are given, while in the second, sides CE = 12 cm and BE = 20 cm are provided. Learn step-by-step how to apply the theorem.

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