The sides of a triangle are in the ratio `(1)/(2) : (1)/(3) : (1)/(4)`. If the perimeter of the triangle is 52 cm, the length of the smallest side is

Options

A. 24 cm
B. 10 cm
C. 12 cm
D. 9 cm

Answer

To find the length of the smallest side of the triangle given the sides in the ratio \( \frac{1}{2} : \frac{1}{3} : \frac{1}{4} \) and a perimeter of 52 cm, follow these steps:

### Step 1: Identify the ratios
The sides of the triangle are given in the ratio \( \frac{1}{2} : \frac{1}{3} : \frac{1}{4} \).

### Step 2: Find the LCM of the denominators
The denominators are 2, 3, and 4. The least common multiple (LCM) of these numbers is 12.

### Step 3: Convert the ratios to whole numbers
Multiply each part of the ratio by the LCM (12):
- For \( \frac{1}{2} \): \( 12 \times \frac{1}{2} = 6 \)
- For \( \frac{1}{3} \): \( 12 \times \frac{1}{3} = 4 \)
- For \( \frac{1}{4} \): \( 12 \times \frac{1}{4} = 3 \)

Thus, the sides of the triangle can be represented as 6, 4, and 3.

### Step 4: Calculate the sum of the sides
Now, add these values to find the total length of the sides:
\[ 6 + 4 + 3 = 13 \]

### Step 5: Set up a proportion to find the actual side lengths
The perimeter of the triangle is given as 52 cm. We can set up a proportion:
\[ \text{Perimeter ratio} = \frac{\text{Actual perimeter}}{\text{Ratio perimeter}} = \frac{52}{13} = 4 \]

### Step 6: Find the actual lengths of the sides
Now, multiply each side of the ratio by this factor (4):
- The first side: \( 6 \times 4 = 24 \) cm
- The second side: \( 4 \times 4 = 16 \) cm
- The third side: \( 3 \times 4 = 12 \) cm

### Step 7: Identify the smallest side
Among the sides 24 cm, 16 cm, and 12 cm, the smallest side is 12 cm.

### Conclusion
The length of the smallest side is **12 cm**.

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