`ΔABC and ΔPQR` are two similar triangles and their respective areas are `81 cm^(2) and 64 cm^(2)` . If the length of the median of `ΔABC` is 10.8 cm then find the length of the median of `ΔPQR`.

Options

A. (A). 10.6cm
B. (B). 9.6cm
C. (C). 8.53cm
D. (D). 12.15cm

Answer

To solve the problem, we need to find the length of the median of triangle \( \Delta PQR \) given that \( \Delta ABC \) and \( \Delta PQR \) are similar triangles with respective areas of \( 81 \, \text{cm}^2 \) and \( 64 \, \text{cm}^2 \). The length of the median of triangle \( \Delta ABC \) is \( 10.8 \, \text{cm} \).

### Step-by-Step Solution:

1. **Understanding the Ratio of Areas**:
Since the triangles are similar, the ratio of their areas is equal to the square of the ratio of their corresponding sides. Let the ratio of the sides of \( \Delta ABC \) to \( \Delta PQR \) be \( k \). Thus, we have:
\[ \frac{\text{Area of } \Delta ABC}{\text{Area of } \Delta PQR} = \frac{81}{64} = \left(\frac{AB}{PQ}\right)^2 = k^2 \]

2. **Finding the Ratio of Sides**:
To find \( k \), we take the square root of the area ratio:
\[ k = \sqrt{\frac{81}{64}} = \frac{9}{8} \]

3. **Using the Ratio of Medians**:
The ratio of the medians of similar triangles is the same as the ratio of their corresponding sides. Therefore, if \( AD \) is the median of \( \Delta ABC \) and \( PS \) is the median of \( \Delta PQR \), we can write:
\[ \frac{AD}{PS} = \frac{9}{8} \]

4. **Substituting the Known Value**:
We know that \( AD = 10.8 \, \text{cm} \). Substituting this value into the equation gives:
\[ \frac{10.8}{PS} = \frac{9}{8} \]

5. **Cross-Multiplying to Solve for \( PS \)**:
Cross-multiplying to find \( PS \):
\[ 10.8 \cdot 8 = 9 \cdot PS \]
\[ 86.4 = 9 \cdot PS \]

6. **Calculating \( PS \)**:
Now, divide both sides by 9 to find \( PS \):
\[ PS = \frac{86.4}{9} = 9.6 \, \text{cm} \]

### Final Answer:
The length of the median of triangle \( \Delta PQR \) is \( 9.6 \, \text{cm} \).

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