If `DeltaABC~DeltaPQR` then evaluate the length of AC. If perimeter of `DeltaABC = 20 cm,` perimeter of `DeltaPQR = 40` cm and PR = 8 cm.

Options

A. 4 cm
B. 6 cm
C. 10 cm
D. 3 cm

Answer

To find the length of AC in triangle ABC, given that triangle ABC is similar to triangle PQR, we can use the properties of similar triangles and the relationship between their perimeters.

### Step-by-Step Solution:

1. **Understand the Relationship of Similar Triangles**:
Since triangle ABC is similar to triangle PQR, the ratios of corresponding sides are equal. This means:
\[ \frac{AB}{PQ} = \frac{AC}{PR} = \frac{BC}{QR} \]

2. **Set Up the Ratio of Perimeters**:
The ratio of the perimeters of the two triangles is equal to the ratio of their corresponding sides. Given:
- Perimeter of triangle ABC = 20 cm
- Perimeter of triangle PQR = 40 cm

Therefore, the ratio of the perimeters is:
\[ \frac{\text{Perimeter of } \Delta ABC}{\text{Perimeter of } \Delta PQR} = \frac{20}{40} = \frac{1}{2} \]

3. **Use the Ratio to Find AC**:
We know that PR corresponds to AC in the similar triangles. Given that PR = 8 cm, we can set up the equation:
\[ \frac{AC}{PR} = \frac{1}{2} \]
Substituting the value of PR:
\[ \frac{AC}{8} = \frac{1}{2} \]

4. **Solve for AC**:
To find AC, we can cross-multiply:
\[ AC = 8 \times \frac{1}{2} = 4 \text{ cm} \]

### Final Answer:
The length of AC is **4 cm**.

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