`Delta ABC` is similar to `Delta PQR`. Length of AB is 16 cm and length of the corresponding side PQ is 9 cm. If area of `Delta ABC` is 1024 sq. cm, what is the area of `Delta PQR `?

Options

A. 768 sq. cm
B. 32 sq. cm
C. 324 sq. cm
D. 128 sq. cm

Answer

To solve the problem, we will follow these steps:

### Step 1: Identify the given information
We know that:
- Triangle ABC is similar to Triangle PQR.
- Length of side AB = 16 cm
- Length of corresponding side PQ = 9 cm
- Area of Triangle ABC = 1024 sq. cm

### Step 2: Calculate the ratio of the sides
The ratio of the corresponding sides of the triangles is given by:
\[ \text{Ratio of sides} = \frac{AB}{PQ} = \frac{16}{9} \]

### Step 3: Use the property of similar triangles
For similar triangles, the ratio of their areas is equal to the square of the ratio of their corresponding sides. Therefore, we can write:
\[ \frac{\text{Area of } \Delta ABC}{\text{Area of } \Delta PQR} = \left(\frac{AB}{PQ}\right)^2 \]
Substituting the known values:
\[ \frac{1024}{\text{Area of } \Delta PQR} = \left(\frac{16}{9}\right)^2 \]

### Step 4: Calculate the square of the ratio of sides
Calculating the square of the ratio:
\[ \left(\frac{16}{9}\right)^2 = \frac{256}{81} \]

### Step 5: Set up the equation
Now we can set up the equation:
\[ \frac{1024}{\text{Area of } \Delta PQR} = \frac{256}{81} \]

### Step 6: Cross-multiply to solve for the area of Triangle PQR
Cross-multiplying gives:
\[ 1024 \cdot 81 = 256 \cdot \text{Area of } \Delta PQR \]

### Step 7: Calculate the area of Triangle PQR
Now, we can solve for the area of Triangle PQR:
\[ \text{Area of } \Delta PQR = \frac{1024 \cdot 81}{256} \]

### Step 8: Simplify the calculation
Calculating the right-hand side:
\[ \text{Area of } \Delta PQR = \frac{1024 \cdot 81}{256} = \frac{1024}{256} \cdot 81 = 4 \cdot 81 = 324 \text{ sq. cm} \]

### Final Answer
The area of Triangle PQR is \( 324 \text{ sq. cm} \).

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