If perimeter of a triangle is 24cm and sides are in the ration 2:1:3, then find the longest side?

Answer

To find the longest side of a triangle given its perimeter and the ratio of its sides, we can follow these steps:

### Step-by-Step Solution:

1. **Understand the Given Information:**
- The perimeter of the triangle is 24 cm.
- The sides are in the ratio 2:1:3.

2. **Assign Variables to the Sides:**
- Let the sides of the triangle be represented as:
- First side (A) = 2k
- Second side (B) = k
- Third side (C) = 3k
- Here, \( k \) is a common multiplier.

3. **Set Up the Equation for the Perimeter:**
- The perimeter of the triangle is the sum of its sides:
\[ A + B + C = 24 \text{ cm} \]
- Substituting the values of A, B, and C:
\[ 2k + k + 3k = 24 \]

4. **Combine Like Terms:**
- Combine the terms on the left side:
\[ 6k = 24 \]

5. **Solve for k:**
- Divide both sides by 6 to find the value of \( k \):
\[ k = \frac{24}{6} = 4 \text{ cm} \]

6. **Calculate the Lengths of the Sides:**
- Now substitute \( k \) back into the expressions for the sides:
- First side (A) = \( 2k = 2 \times 4 = 8 \text{ cm} \)
- Second side (B) = \( k = 4 \text{ cm} \)
- Third side (C) = \( 3k = 3 \times 4 = 12 \text{ cm} \)

7. **Identify the Longest Side:**
- Compare the lengths of the sides:
- A = 8 cm
- B = 4 cm
- C = 12 cm
- The longest side is \( C \), which is 12 cm.

### Final Answer:
The longest side of the triangle is **12 cm**.