In ΔABC, AB = 4 and BC = 7, Then , .... holds good.

Options

A. `AC lt7`
B. `AC gt4`
C. `4 lt AC lt 7`
D. `3 lt AC lt 11`

Answer

To solve the problem, we need to determine the possible range for the length of side AC in triangle ABC, given that AB = 4 and BC = 7. We will use the triangle inequality theorem, which states that the sum of the lengths of any two sides of a triangle must be greater than the length of the third side.

### Step-by-Step Solution:

1. **Identify the sides of the triangle**: We have the lengths of two sides, AB = 4 and BC = 7. We need to find the possible range for the third side, AC.

2. **Apply the triangle inequality theorem**: According to the triangle inequality theorem, the following inequalities must hold true for triangle ABC:
- AB + AC > BC
- AB + BC > AC
- AC + BC > AB

3. **Set up the inequalities**:
- From the first inequality:
\[ 4 + AC > 7 \implies AC > 7 - 4 \implies AC > 3 \]
- From the second inequality:
\[ 4 + 7 > AC \implies 11 > AC \implies AC < 11 \]
- The third inequality is:
\[ AC + 7 > 4 \implies AC > 4 - 7 \implies AC > -3 \]
(This inequality does not provide a new constraint since AC must be positive.)

4. **Combine the inequalities**:
From the first two inequalities, we have:
\[ 3 < AC < 11 \]

5. **Conclusion**: The length of side AC must be greater than 3 and less than 11. Therefore, we can conclude:
\[ AC \text{ lies in the range } (3, 11). \]

### Final Answer:
The condition that holds good is that AC lies between 3 and 11.

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