The coordinates of the endpoints of 
PQ
 are P(3,5) and Q(18,15). Point R is on 
PQ
 and divides it such that PR:QR is 1:4.
What are the coordinates of R?
Write your answers as integers or decimals.

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.

The coordinates of the endpoints of PQ are P(3,5) and Q(18,15). Point R is on PQ and divides it such that PR:QR is 1:4. What are the coordinates of R? Write your answers as integers or decimals.

Learn how to find the coordinates of a point that divides a line segment in a specific ratio using the section formula. This problem involves coordinate geometry and ratio concepts, ideal for students in Grades 8-10. The given points are P(3,5) and Q(18,15), with the ratio PR:QR = 1:4.

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