Let `triangleABC ~ triangleQPR` and `(ar(triangleABC))/(ar(trianglePQR))=9/16`. If AB=12 cm, BC=6 cm and AC=9 cm then QR is equal to :

Options

A. 12 cm
B. 16 cm
C. 8 cm
D. 9 cm

Answer

To solve the problem, we need to find the length of side QR in triangle QPR, given that triangles ABC and QPR are similar and the ratio of their areas is 9/16.

### Step-by-Step Solution:

1. **Understanding the Similarity of Triangles**:
Since triangle ABC is similar to triangle QPR, the ratios of their corresponding sides are equal. This means:
\[ \frac{AB}{QP} = \frac{BC}{PR} = \frac{AC}{QR} \]

2. **Using the Area Ratio**:
The ratio of the areas of two similar triangles is equal to the square of the ratio of their corresponding sides. Given:
\[ \frac{ar(\triangle ABC)}{ar(\triangle QPR)} = \frac{9}{16} \]
This implies:
\[ \left(\frac{AB}{QP}\right)^2 = \frac{9}{16} \]

3. **Finding the Ratio of Corresponding Sides**:
Taking the square root of both sides gives us the ratio of the corresponding sides:
\[ \frac{AB}{QP} = \frac{3}{4} \]

4. **Substituting the Known Value of AB**:
We know that \( AB = 12 \) cm. Therefore, we can set up the equation:
\[ \frac{12}{QP} = \frac{3}{4} \]

5. **Cross-Multiplying to Solve for QP**:
Cross-multiplying gives:
\[ 3 \cdot QP = 12 \cdot 4 \]
\[ 3 \cdot QP = 48 \]
\[ QP = \frac{48}{3} = 16 \text{ cm} \]

6. **Finding the Length of QR**:
Now we can use the ratio of sides to find QR. We know:
\[ \frac{AC}{QR} = \frac{3}{4} \]
Substituting \( AC = 9 \) cm:
\[ \frac{9}{QR} = \frac{3}{4} \]

7. **Cross-Multiplying to Solve for QR**:
Cross-multiplying gives:
\[ 3 \cdot QR = 9 \cdot 4 \]
\[ 3 \cdot QR = 36 \]
\[ QR = \frac{36}{3} = 12 \text{ cm} \]

### Final Answer:
QR is equal to **12 cm**.