To find the coordinates of point $R$ that divides the line segment $PQ$ in the ratio 1:4, we can use the section formula. The section formula gives the coordinates of a point dividing a line segment in a given ratio.

The formula for the coordinates $(x, y)$ of the point dividing a line segment joining two points $P(x_1, y_1)$ and $Q(x_2, y_2)$ in the ratio $m:n$ is:

$(x, y) = \left( \frac{mx_2 + nx_1}{m+n}, \frac{my_2 + ny_1}{m+n} \right)$

Given:

• $P(3, 5)$
• $Q(18, 15)$
• Ratio $PR:QR = 1:4$, so $m = 1$ and $n = 4$.

Using the section formula, we can now calculate the coordinates of point $R$.

Step 1: Calculate the x-coordinate of $R$

$x_R = \frac{(1 \times 18) + (4 \times 3)}{1+4} = \frac{18 + 12}{5} = \frac{30}{5} = 6$

Step 2: Calculate the y-coordinate of $R$

$y_R = \frac{(1 \times 15) + (4 \times 5)}{1+4} = \frac{15 + 20}{5} = \frac{35}{5} = 7$

Thus, the coordinates of point $R$ are $\boxed{(6, 7)}$.

Would you like any more details or explanations? Here are some related questions:

1. How does the section formula change for different ratios?
2. Can the section formula be applied in three dimensions?
3. What if $PR:QR$ was 4:1 instead of 1:4?
4. How would you find the midpoint of a line segment?
5. How can you determine if point $R$ lies on the segment $PQ$?

Tip: The section formula can be used in both internal and external divisions of a line segment.