Two sides of an isosceles triangle are 4 cm and 7. what is the possible measure of the third side ?

Answer

To find the possible measure of the third side of an isosceles triangle where two sides are given as 4 cm and 7 cm, we can follow these steps:

### Step-by-Step Solution:

1. **Identify the Properties of an Isosceles Triangle**:
An isosceles triangle has at least two sides that are equal in length. In this case, we have two sides measuring 4 cm and 7 cm.

2. **Determine Which Side is Equal**:
Since we have two different lengths (4 cm and 7 cm), one of these lengths must be the length of the two equal sides. Therefore, we can consider two cases:
- Case 1: The two equal sides are both 4 cm.
- Case 2: The two equal sides are both 7 cm.

3. **Apply 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 will apply this theorem to both cases.

**Case 1**: If the two equal sides are 4 cm:
- Let the third side be \( x \).
- According to the triangle inequality:
- \( 4 + 4 > x \) → \( 8 > x \) → \( x < 8 \)
- \( 4 + x > 4 \) → \( x > 0 \) (This condition is always true for positive lengths)
- \( 4 + x > 4 \) → \( x > 0 \) (This condition is also always true)

From Case 1, we find that \( x < 8 \).

**Case 2**: If the two equal sides are 7 cm:
- Let the third side be \( x \).
- According to the triangle inequality:
- \( 7 + 7 > x \) → \( 14 > x \) → \( x < 14 \)
- \( 7 + x > 7 \) → \( x > 0 \) (This condition is always true)
- \( 7 + x > 7 \) → \( x > 0 \) (This condition is also always true)

From Case 2, we find that \( x < 14 \).

4. **Combine the Results**:
Now we need to find the possible values for \( x \) based on both cases:
- From Case 1, \( x < 8 \)
- From Case 2, \( x < 14 \)

However, since we need to consider the maximum possible length of the third side based on the sides we have, we also need to consider the minimum side length:
- The third side must be greater than the absolute difference of the two sides, which is \( |7 - 4| = 3 \).

5. **Final Range for the Third Side**:
Therefore, combining these results, the possible measure of the third side \( x \) must satisfy:
- \( 3 < x < 8 \)

### Conclusion:
The possible measure of the third side of the isosceles triangle can be any value greater than 3 cm and less than 8 cm.