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### Problem Statement
The image shows two triangles. The left triangle is a right-angled triangle with sides labeled as follows:
- The side opposite the right angle is labeled \(c = 17\) cm.
- One leg is labeled \(a = 15\) cm.
- The other leg is labeled \(b\).

The right triangle is a non-right-angled triangle labeled as follows:
- One side is labeled \(c = 10\) cm.
- Another side is labeled \(b = 8\) cm.
- The angle between these sides is labeled \( \alpha = 60^\circ \).

### Task
1. For the left triangle, we need to find the length of the unknown side \(b\).
2. For the right triangle, we need to find the length of the third side using the given angle.

### Step-by-Step Solution

1. **For the Left Triangle (Right-Angled):**

   Since it's a right-angled triangle, we can use the **Pythagorean Theorem**:
   \[
   c^2 = a^2 + b^2
   \]
   Given \(c = 17\) cm and \(a = 15\) cm, we substitute into the equation:
   \[
   17^2 = 15^2 + b^2
   \]
   \[
   289 = 225 + b^2
   \]
   \[
   b^2 = 289 - 225 = 64
   \]
   \[
   b = \sqrt{64} = 8 \text{ cm}
   \]

2. **For the Right Triangle (Non-Right-Angled):**

   We use the **Cosine Rule** to find the unknown side \(a\):
   \[
   a^2 = b^2 + c^2 - 2bc \cdot \cos(\alpha)
   \]
   Given \(b = 8\) cm, \(c = 10\) cm, and \(\alpha = 60^\circ\), we substitute into the equation:
   \[
   a^2 = 8^2 + 10^2 - 2 \cdot 8 \cdot 10 \cdot \cos(60^\circ)
   \]
   \[
   a^2 = 64 + 100 - 2 \cdot 8 \cdot 10 \cdot \left(\frac{1}{2}\right)
   \]
   \[
   a^2 = 64 + 100 - 80
   \]
   \[
   a^2 = 84
   \]
   \[
   a = \sqrt{84} \approx 9.17 \text{ cm}
   \]

### Summary of Answers:
- For the left triangle, \(b = 8\) cm.
- For the right triangle, \(a \approx 9.17\) cm.

Would you like more details or have any questions about this solution? 

### Related Questions
1. How would you find the area of each triangle?
2. What is the perimeter of the right triangle after finding all sides?
3. How does the Cosine Rule change if the angle \(\alpha\) is obtuse?
4. Can you calculate the angles of the left triangle using trigonometric ratios?
5. What happens if the angle \(\alpha\) in the right triangle was given as \(90^\circ\)?

### Tip
Always remember to check whether a triangle is right-angled before choosing the appropriate theorem or formula for solving side lengths.

The image contains two triangle problems: one is a right-angled triangle with known side lengths, and the other is a non-right-angled triangle with a given angle and two sides.

This page provides a detailed solution to problems involving a right-angled triangle and a non-right-angled triangle. Learn how to apply the Pythagorean Theorem and the Cosine Rule to find unknown sides. Suitable for students in Grades 8-10.

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