If `DeltaABC~DeltaQRP,(ar(DeltaABC))/(ar(DeltaPQR))=9/4`,AB=18 cm and BC=15 cm, then PR is equal to

Options

A. 10 cm
B. 12 cm
C. `20/3` cm
D. 8 cm

Explanation

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

### Step-by-Step Solution:

1. **Understanding Similar Triangles**:
Since triangle ABC is similar to triangle QRP, we can use the properties of similar triangles. The ratio of the areas of two similar triangles is equal to the square of the ratio of their corresponding sides.

2. **Setting Up the Ratio of Areas**:
We are given that:
\[ \frac{ar(\Delta ABC)}{ar(\Delta PQR)} = \frac{9}{4} \]
Let the corresponding side of triangle ABC be \( BC \) and the corresponding side of triangle QRP be \( PR \).

3. **Using the Ratio of Areas**:
The ratio of the areas can also be expressed in terms of the sides:
\[ \frac{ar(\Delta ABC)}{ar(\Delta PQR)} = \left(\frac{BC}{PR}\right)^2 \]
Therefore, we can write:
\[ \left(\frac{BC}{PR}\right)^2 = \frac{9}{4} \]

4. **Substituting the Length of BC**:
We know that \( BC = 15 \, \text{cm} \). Substituting this into the equation gives:
\[ \left(\frac{15}{PR}\right)^2 = \frac{9}{4} \]

5. **Taking the Square Root**:
Taking the square root of both sides, we get:
\[ \frac{15}{PR} = \frac{3}{2} \]

6. **Cross Multiplying**:
Cross multiplying gives:
\[ 3 \cdot PR = 2 \cdot 15 \]
Simplifying the right side:
\[ 3 \cdot PR = 30 \]

7. **Solving for PR**:
Dividing both sides by 3 to isolate \( PR \):
\[ PR = \frac{30}{3} = 10 \, \text{cm} \]

### Final Answer:
Thus, the length of \( PR \) is \( 10 \, \text{cm} \).

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