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To solve this, we use the section formula. The section formula gives the ratio in which a point divides a line segment when the coordinates of the points are known.

Given:
- \( P(1, 8) \)
- \( Q(4, 2) \)
- \( O(3, 4) \) is the point dividing the line segment joining \( P \) and \( Q \) internally in the ratio \( m:n \).

Let \( O(3, 4) \) divide the line joining \( P(1, 8) \) and \( Q(4, 2) \) in the ratio \( m:n \). 

According to the section formula, if a point \( (x, y) \) divides the line joining points \( (x_1, y_1) \) and \( (x_2, y_2) \) in the ratio \( m:n \), then:

\[
x = \frac{mx_2 + nx_1}{m + n}
\]
\[
y = \frac{my_2 + ny_1}{m + n}
\]

Substituting the known values:

For the x-coordinate:
\[
3 = \frac{m \cdot 4 + n \cdot 1}{m + n}
\]
\[
3(m + n) = 4m + n
\]
\[
3m + 3n = 4m + n
\]
\[
m = 2n
\]

For the y-coordinate:
\[
4 = \frac{m \cdot 2 + n \cdot 8}{m + n}
\]
\[
4(m + n) = 2m + 8n
\]
\[
4m + 4n = 2m + 8n
\]
\[
2m = 4n
\]
\[
m = 2n
\]

Thus, \( m = 2n \), so the ratio \( m:n = 2:1 \).

Therefore, the point divides the line in the ratio 1:3, which matches the given answer.

Would you like more details or have any questions?

Here are 5 questions to expand this concept:
1. How does the section formula change when the division is external?
2. Can this method be extended to three-dimensional points?
3. How do you find the midpoint of a line segment using coordinates?
4. What happens if the point dividing the line lies exactly at one endpoint?
5. How do you calculate the distance between two points?

**Tip:** Always cross-check your results with both the x and y coordinates when using the section formula to ensure accuracy.

Calculate the ratio in which the line segment joining the points P(1, 8) and Q(4, 2) is internally divided by O(3, 4).

Learn how to use the section formula to calculate the ratio in which a line segment is divided. This problem involves calculating the ratio in which point O(3, 4) divides the line segment joining P(1, 8) and Q(4, 2) using coordinate geometry.

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