The perimeter of a `DeltaABC` is `48 cm` and one side is `20 cm`. Then remaining sides of `DeltaABC` must be greater than : (a) `8 cm` (b) `9 cm` (c) `12 cm` (d) `4 cm`

Options

A. 8 cm
B. 9 cm
C. 12 cm
D. 4 cm

Answer

To solve the problem, we need to find the minimum value of the remaining sides of triangle ABC given that the perimeter is 48 cm and one side is 20 cm.

### Step-by-Step Solution:

1. **Identify the given information**:
- Perimeter of triangle ABC = 48 cm
- One side (let's say side A) = 20 cm

2. **Set up the equation for the perimeter**:
- Let the other two sides be B and C.
- According to the perimeter, we have:
\[ A + B + C = 48 \]
- Substituting A = 20 cm:
\[ 20 + B + C = 48 \]

3. **Solve for B + C**:
- Rearranging the equation gives:
\[ B + C = 48 - 20 = 28 \]

4. **Apply the triangle inequality**:
- The triangle inequality states that the sum of the lengths of any two sides must be greater than the length of the third side. Therefore, we have:
- \( A + B > C \)
- \( A + C > B \)
- \( B + C > A \)

5. **Substituting A = 20 cm into the inequalities**:
- From \( A + B > C \):
\[ 20 + B > C \quad \text{(1)} \]
- From \( A + C > B \):
\[ 20 + C > B \quad \text{(2)} \]
- From \( B + C > A \):
\[ B + C > 20 \quad \text{(3)} \]
- We already know from step 3 that \( B + C = 28 \), so:
\[ 28 > 20 \quad \text{(This is always true)} \]

6. **Rearranging inequalities (1) and (2)**:
- From inequality (1):
\[ C < 20 + B \]
- From inequality (2):
\[ B < 20 + C \]

7. **Express C in terms of B**:
- From \( B + C = 28 \), we can express C as:
\[ C = 28 - B \]
- Substitute this into inequality (1):
\[ 28 - B < 20 + B \]
- Rearranging gives:
\[ 28 - 20 < 2B \implies 8 < 2B \implies B > 4 \]

8. **Express B in terms of C**:
- Similarly, substitute into inequality (2):
\[ B < 20 + (28 - B) \]
- Rearranging gives:
\[ B < 48 - B \implies 2B < 48 \implies B < 24 \]

9. **Conclusion**:
- From the inequalities derived:
- \( B > 4 \)
- \( B < 24 \)
- Since B and C are interchangeable, we can conclude that both remaining sides must be greater than 4 cm.

### Final Answer:
The remaining sides of triangle ABC must be greater than **4 cm**.