Question 9

Dividing AB into four equal parts means we need 3 points — P, Q, R — where each sits at $\frac{1}{4}$, $\frac{2}{4}$, and $\frac{3}{4}$ of the way from A to B. This gives ratios $1:3$, $1:1$, and $3:1$ respectively.

Section Formula: If a point $\text{P}(x, y)$ divides the line segment joining $\text{A}(x_1, y_1)$ and $\text{B}(x_2, y_2)$ in the ratio $m:n$ internally, then:

$x = \frac{mx_2 + nx_1}{m + n}, \quad y = \frac{my_2 + ny_1}{m + n}$

We will use the section formula to find the coordinates of the points.

Step 1 — Find the first point (P)

Let the points be P, Q, and R. They divide the line segment AB into four equal parts. Point P is the first point. P divides AB in the ratio 1:3. Let A be $(x_1, y_1) = \mathbf{(-2, 2)}$. Let B be $(x_2, y_2) = \mathbf{(2, 8)}$. The ratio $m:n$ is $\mathbf{1:3}$. We use the section formula for the x-coordinate:

$P_x = \frac{nx_1 + mx_2}{m+n}$

$P_x = \frac{3(-2) + 1(2)}{1+3}$

$P_x = \frac{-6 + 2}{4}$

$P_x = \frac{-4}{4}$

$P_x = -1$ Now, we find the y-coordinate:

$P_y = \frac{ny_1 + my_2}{m+n}$

$P_y = \frac{3(2) + 1(8)}{1+3}$

$P_y = \frac{6 + 8}{4}$

$P_y = \frac{14}{4}$

$\boxed{P_y = \frac{7}{2}}$

Step 2 — Find the second point (Q)

Point Q is the midpoint of AB. It divides AB in the ratio 1:1. We use the midpoint formula. Let A be $(x_1, y_1) = \mathbf{(-2, 2)}$. Let B be $(x_2, y_2) = \mathbf{(2, 8)}$. We find the x-coordinate of Q:

$Q_x = \frac{x_1+x_2}{2}$

$Q_x = \frac{-2+2}{2}$

$Q_x = \frac{0}{2}$

$Q_x = 0$ Now, we find the y-coordinate of Q:

$Q_y = \frac{y_1+y_2}{2}$

$Q_y = \frac{2+8}{2}$

$Q_y = \frac{10}{2}$

$\boxed{Q_y = 5}$

Step 3 — Find the third point (R)

Point R is the third point. It divides AB in the ratio 3:1. Let A be $(x_1, y_1) = \mathbf{(-2, 2)}$. Let B be $(x_2, y_2) = \mathbf{(2, 8)}$. The ratio $m:n$ is $\mathbf{3:1}$. We use the section formula for the x-coordinate:

$R_x = \frac{nx_1 + mx_2}{m+n}$

$R_x = \frac{1(-2) + 3(2)}{3+1}$

$R_x = \frac{-2 + 6}{4}$

$R_x = \frac{4}{4}$

$R_x = 1$ Now, we find the y-coordinate:

$R_y = \frac{ny_1 + my_2}{m+n}$

$R_y = \frac{1(2) + 3(8)}{3+1}$

$R_y = \frac{2 + 24}{4}$

$R_y = \frac{26}{4}$

$\boxed{R_y = \frac{13}{2}}$

Answer

(i) The coordinates of the first point are $\mathbf{(-1, \frac{7}{2})}$. (ii) The coordinates of the second point are $\mathbf{(0, 5)}$. (iii) The coordinates of the third point are $\mathbf{(1, \frac{13}{2})}$.