The area of `ΔABC and ΔPQR` is `144 cm^(2) and 49 cm^(2)` respectively, if the length of the side AC is 9 cm, then find PR.

Options

A. 4.5 cm
B. 5.25 cm
C. 5.5 cm
D. 4.25 cm

Answer

To solve the problem, we will use the properties of similar triangles and the relationship between the areas of the triangles and the lengths of their corresponding sides.

### Step-by-Step Solution:

1. **Identify the Areas of the Triangles**:
- Area of triangle ABC = 144 cm²
- Area of triangle PQR = 49 cm²

2. **Identify the Length of Side AC**:
- Length of side AC = 9 cm

3. **Use the Property of Similar Triangles**:
Since triangles ABC and PQR are similar, the ratio of their areas is equal to the square of the ratio of their corresponding sides. Therefore, we can write:
\[ \frac{\text{Area of } \Delta ABC}{\text{Area of } \Delta PQR} = \left(\frac{AC}{PR}\right)^2 \]

4. **Substitute the Known Values**:
\[ \frac{144}{49} = \left(\frac{9}{PR}\right)^2 \]

5. **Take the Square Root of Both Sides**:
\[ \sqrt{\frac{144}{49}} = \frac{9}{PR} \]
This simplifies to:
\[ \frac{12}{7} = \frac{9}{PR} \]

6. **Cross-Multiply to Solve for PR**:
\[ 12 \cdot PR = 9 \cdot 7 \]
\[ 12 \cdot PR = 63 \]

7. **Divide Both Sides by 12**:
\[ PR = \frac{63}{12} \]
Simplifying this gives:
\[ PR = 5.25 \text{ cm} \]

### Final Answer:
The length of side PR is **5.25 cm**.