To find the points that divide the line segment joining the points (8, 12) and (12, 8) into four equal parts, we will follow these steps:
### Step 1: Identify the Points
The two endpoints of the line segment are:
- A(8, 12)
- B(12, 8)
### Step 2: Determine the Ratio for Division
Since we want to divide the line segment into four equal parts, we will have three points (let's call them P, Q, and R) that divide the segment in the ratios:
- P divides AB in the ratio 1:3
- Q divides AB in the ratio 1:1 (midpoint)
- R divides AB in the ratio 3:1
### Step 3: Calculate the Coordinates of Point P
Using the section formula, the coordinates of point P which divides the segment in the ratio 1:3 can be calculated as follows:
\[
P(x, y) = \left( \frac{m x_2 + n x_1}{m+n}, \frac{m y_2 + n y_1}{m+n} \right)
\]
Where:
- \( (x_1, y_1) = (8, 12) \)
- \( (x_2, y_2) = (12, 8) \)
- \( m = 1 \) (part towards B)
- \( n = 3 \) (part towards A)
Calculating the x-coordinate of P:
\[
x_P = \frac{1 \cdot 12 + 3 \cdot 8}{1 + 3} = \frac{12 + 24}{4} = \frac{36}{4} = 9
\]
Calculating the y-coordinate of P:
\[
y_P = \frac{1 \cdot 8 + 3 \cdot 12}{1 + 3} = \frac{8 + 36}{4} = \frac{44}{4} = 11
\]
Thus, the coordinates of point P are \( P(9, 11) \).
### Step 4: Calculate the Coordinates of Point Q
Point Q is the midpoint of the line segment, which divides it in the ratio 1:1.
Using the midpoint formula:
\[
Q(x, y) = \left( \frac{x_1 + x_2}{2}, \frac{y_1 + y_2}{2} \right)
\]
Calculating the coordinates of Q:
\[
x_Q = \frac{8 + 12}{2} = \frac{20}{2} = 10
\]
\[
y_Q = \frac{12 + 8}{2} = \frac{20}{2} = 10
\]
Thus, the coordinates of point Q are \( Q(10, 10) \).
### Step 5: Calculate the Coordinates of Point R
Using the section formula again, point R divides the segment in the ratio 3:1.
Calculating the x-coordinate of R:
\[
x_R = \frac{3 \cdot 12 + 1 \cdot 8}{3 + 1} = \frac{36 + 8}{4} = \frac{44}{4} = 11
\]
Calculating the y-coordinate of R:
\[
y_R = \frac{3 \cdot 8 + 1 \cdot 12}{3 + 1} = \frac{24 + 12}{4} = \frac{36}{4} = 9
\]
Thus, the coordinates of point R are \( R(11, 9) \).
### Final Answer
The points that divide the line segment joining (8, 12) and (12, 8) into four equal parts are:
- P(9, 11)
- Q(10, 10)
- R(11, 9)
