Given that : `Delta ABC ~ DeltaPQR` , If `("area " (Delta PQR))/("area " (Delta ABC)) = (256)/(441)` and PR = 12 cm, then AC is equal to

Options

A. 15.75 cm
B. 16 cm
C. `12sqrt(2) cm`
D. 15.5 cm

Answer

To find the length of AC given that triangle ABC is similar to triangle PQR, we can use the relationship between the areas of similar triangles and the lengths of their corresponding sides.

### Step-by-Step Solution:

1. **Understanding the Relationship Between Areas and Sides**:
Since triangles ABC and PQR are similar, the ratio of their areas is equal to the square of the ratio of their corresponding sides.
\[ \frac{\text{Area of } \Delta PQR}{\text{Area of } \Delta ABC} = \left(\frac{PR}{AC}\right)^2 \]

2. **Substituting the Given Values**:
We know:
\[ \frac{\text{Area of } \Delta PQR}{\text{Area of } \Delta ABC} = \frac{256}{441} \]
and \( PR = 12 \) cm. We can substitute these values into the equation:
\[ \frac{256}{441} = \left(\frac{12}{AC}\right)^2 \]

3. **Cross-Multiplying to Solve for AC**:
To eliminate the fraction, we cross-multiply:
\[ 256 \cdot AC^2 = 441 \cdot 12^2 \]

4. **Calculating \( 12^2 \)**:
First, calculate \( 12^2 \):
\[ 12^2 = 144 \]
Now substitute this back into the equation:
\[ 256 \cdot AC^2 = 441 \cdot 144 \]

5. **Calculating \( 441 \cdot 144 \)**:
Now, calculate \( 441 \cdot 144 \):
\[ 441 \cdot 144 = 63504 \]
So, we have:
\[ 256 \cdot AC^2 = 63504 \]

6. **Dividing Both Sides by 256**:
Now, divide both sides by 256 to isolate \( AC^2 \):
\[ AC^2 = \frac{63504}{256} \]

7. **Calculating \( \frac{63504}{256} \)**:
Performing the division:
\[ AC^2 = 248.5 \]

8. **Taking the Square Root**:
Finally, take the square root to find AC:
\[ AC = \sqrt{248.5} \approx 15.75 \text{ cm} \]

### Final Answer:
Thus, the length of AC is approximately **15.75 cm**.

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