If corresponding sides of two similar triangles are in the ratio 4 : 5 , then corresponding medians of the triangles are in the ratio `4 : k` . Find K .

Answer

To find the value of \( k \) in the ratio of corresponding medians of two similar triangles, follow these steps:

### Step-by-Step Solution:

1. **Understand the Problem**: We have two similar triangles, say triangle ABC and triangle PQR, with corresponding sides in the ratio \( 4:5 \). We need to find the ratio of their corresponding medians, which is given as \( 4:k \).

2. **Set Up the Ratios**: Since the triangles are similar, the ratio of their corresponding sides is given as:
\[ \frac{AB}{PQ} = \frac{BC}{QR} = \frac{AC}{PR} = \frac{4}{5} \]

3. **Use the Property of Medians**: The ratio of the medians of two similar triangles is equal to the ratio of their corresponding sides. Therefore, if the sides are in the ratio \( 4:5 \), the medians will also be in the same ratio:
\[ \frac{m_1}{m_2} = \frac{4}{5} \]
where \( m_1 \) and \( m_2 \) are the lengths of the medians of triangles ABC and PQR, respectively.

4. **Express the Medians in Terms of k**: From the problem, we know that the ratio of the medians is also given as:
\[ \frac{m_1}{m_2} = \frac{4}{k} \]

5. **Set the Two Ratios Equal**: Since both expressions represent the same ratio of medians, we can set them equal to each other:
\[ \frac{4}{5} = \frac{4}{k} \]

6. **Cross Multiply to Solve for k**: Cross multiplying gives us:
\[ 4k = 4 \times 5 \]
Simplifying this, we get:
\[ 4k = 20 \]

7. **Solve for k**: Dividing both sides by 4:
\[ k = \frac{20}{4} = 5 \]

### Final Answer:
Thus, the value of \( k \) is \( 5 \).