Consider two similar triangles ABC and LMN, whose perimeters are respectively 60 cm and 48 cm. If the length of LM is 8 cm, the length of AB is:

Options

A. 10 cm
B. 6 cm
C. 12 cm
D. 14 cm

Answer

To find the length of AB in triangle ABC given that triangles ABC and LMN are similar, we can follow these steps:

### Step 1: Determine the ratio of the perimeters of the triangles.
The perimeters of triangles ABC and LMN are given as:
- Perimeter of triangle ABC = 60 cm
- Perimeter of triangle LMN = 48 cm

The ratio of the perimeters is calculated as:
\[ \text{Ratio of perimeters} = \frac{\text{Perimeter of ABC}}{\text{Perimeter of LMN}} = \frac{60}{48} \]

### Step 2: Simplify the ratio.
To simplify the ratio:
\[ \frac{60}{48} = \frac{60 \div 12}{48 \div 12} = \frac{5}{4} \]

### Step 3: Use the ratio to find the length of AB.
Since the triangles are similar, the ratio of corresponding sides is equal to the ratio of their perimeters. We know the length of side LM in triangle LMN is 8 cm. We need to find the length of side AB in triangle ABC.

Using the ratio:
\[ \frac{AB}{LM} = \frac{5}{4} \]
Substituting the known value of LM:
\[ \frac{AB}{8} = \frac{5}{4} \]

### Step 4: Cross-multiply to solve for AB.
Cross-multiplying gives:
\[ 4 \cdot AB = 5 \cdot 8 \]
\[ 4 \cdot AB = 40 \]

### Step 5: Solve for AB.
Now, divide both sides by 4:
\[ AB = \frac{40}{4} = 10 \text{ cm} \]

### Final Answer:
The length of AB is 10 cm.

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