Let `DeltaABC~DeltaPQR and (ar (DeltaABC))/(ar (DeltaPQR))=4/(25)`. If `AB=12cm, BC = 8cm and AC=10cm` then QR is equal to

Options

A. a)15
B. b)18
C. c)25
D. d)20

Answer

To solve the problem, we need to find the length of side QR in triangle PQR, given that triangle ABC is similar to triangle PQR and the ratio of their areas is 4/25. We also know the lengths of the sides of triangle ABC.

### Step-by-Step Solution:

1. **Understanding 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.

2. **Setting Up the Ratio of Areas**: We are given that:
\[ \frac{ar(\Delta ABC)}{ar(\Delta PQR)} = \frac{4}{25} \]
This implies:
\[ \frac{AB^2}{PQ^2} = \frac{4}{25} \]
where AB corresponds to PQ, BC corresponds to QR, and AC corresponds to RP.

3. **Using the Given Side Lengths**: We know the lengths of the sides of triangle ABC:
- AB = 12 cm
- BC = 8 cm
- AC = 10 cm

4. **Finding the Corresponding Side QR**: We will use the sides BC and QR. The ratio of the areas gives us:
\[ \frac{BC^2}{QR^2} = \frac{4}{25} \]
Substituting BC = 8 cm:
\[ \frac{8^2}{QR^2} = \frac{4}{25} \]

5. **Calculating QR**:
- First, calculate \(8^2\):
\[ 8^2 = 64 \]
- Now, we have:
\[ \frac{64}{QR^2} = \frac{4}{25} \]
- Cross-multiplying gives:
\[ 64 \cdot 25 = 4 \cdot QR^2 \]
- This simplifies to:
\[ 1600 = 4 \cdot QR^2 \]
- Dividing both sides by 4:
\[ QR^2 = 400 \]
- Taking the square root:
\[ QR = \sqrt{400} = 20 \text{ cm} \]

### Final Answer:
The length of QR is **20 cm**.