The lengths of two sides of an isosceles triangle are 5 cm and 12 cm. The length of the third side is _______

Options

A. 12cm
B. 5 cm
C. 17cm
D. 10cm

Explanation

To find the length of the third side of an isosceles triangle where the lengths of the two sides are given as 5 cm and 12 cm, we can follow these steps:

### Step-by-Step Solution:

1. **Identify the properties of an isosceles triangle**: In an isosceles triangle, two sides are of equal length. Therefore, we need to determine which of the given lengths (5 cm or 12 cm) represents the equal sides.

2. **Assume the equal sides**: Let's assume the two equal sides are 5 cm each. This means the third side will be 12 cm.

3. **Check the triangle inequality theorem**: The triangle inequality theorem states that the sum of the lengths of any two sides of a triangle must be greater than the length of the third side. We need to check if this holds true for our assumed lengths:
- 5 cm + 5 cm > 12 cm (10 cm > 12 cm) - This is false.
- Therefore, our assumption that the two equal sides are 5 cm is incorrect.

4. **Reassume the equal sides**: Now, let's assume the two equal sides are 12 cm each. This means the third side will be 5 cm.

5. **Check the triangle inequality theorem again**:
- 12 cm + 12 cm > 5 cm (24 cm > 5 cm) - This is true.
- 12 cm + 5 cm > 12 cm (17 cm > 12 cm) - This is true.
- 12 cm + 5 cm > 12 cm (17 cm > 12 cm) - This is true.

6. **Conclusion**: Since all conditions of the triangle inequality theorem are satisfied, the length of the third side of the isosceles triangle is 5 cm.

### Final Answer:
The length of the third side is **5 cm**.

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