Coordinate Geometry | Exercise 7.2

Question 9

Find the coordinates of the points which divide the line segment joining $A(-2, 2)$ and $B(2, 8)$ into four equal parts.

Solution

Understand the Question

Dividing a line segment $AB$ into four equal parts requires three points: $P$ , $Q$ , and $R$ .
These points divide the segment $AB$ $A B$ internally in the following ratios:
- $P$ divides $AB$ in the ratio $1:3$
- $Q$ is the midpoint of $AB$ (ratio $1:1$ )
- $R$ divides $AB$ in the ratio $3:1$
Section Formula: If a point $(x, y)$ divides the segment joining $(x_1, y_1)$ and $(x_2, y_2)$ in the ratio $m:n$ , then: $x = \dfrac{mx_2 + nx_1}{m+n}, \quad y = \dfrac{my_2 + ny_1}{m+n}$

Step 1 · Find the Coordinates of Point P

Let $P$ , $Q$ , and $R$ divide the line segment joining $A(-2, 2)$ and $B(2, 8)$ into four equal parts.Point $P$ divides $AB$ in the ratio $m:n = 1:3$ .

Using the section formula for $P(x, y)$ :

\begin{aligned} P_x &= \dfrac{nx_1 + mx_2}{m+n} \\[0.6em] &= \dfrac{3(-2) + 1(2)}{1+3} \\[0.6em] &= \dfrac{-6 + 2}{4} \\[0.6em] &= \dfrac{-4}{4} \\[0.6em] &= -1 \end{aligned}

\begin{aligned} P_y &= \dfrac{ny_1 + my_2}{m+n} \\[0.6em] &= \dfrac{3(2) + 1(8)}{1+3} \\[0.6em] &= \dfrac{6 + 8}{4} \\[0.6em] &= \dfrac{14}{4} \\[0.6em] &= \dfrac{7}{2} \end{aligned}

Thus, $P = \left(-1, \dfrac{7}{2}\right)$ .

Step 2 · Find the Coordinates of Point Q

Point $Q$ is the midpoint of $AB$ , dividing it in the ratio $1:1$ .

Using the midpoint formula:

\begin{aligned} Q_x &= \dfrac{x_1+x_2}{2} \\[0.6em] &= \dfrac{-2+2}{2} \\[0.6em] &= \dfrac{0}{2} \\[0.6em] &= 0 \end{aligned}

\begin{aligned} Q_y &= \dfrac{y_1+y_2}{2} \\[0.6em] &= \dfrac{2+8}{2} \\[0.6em] &= \dfrac{10}{2} \\[0.6em] &= 5 \end{aligned}

Thus, $Q = (0, 5)$ .

Step 3 · Find the Coordinates of Point R

Point $R$ divides $AB$ in the ratio $m:n = 3:1$ .

Using the section formula for $R(x, y)$ :

\begin{aligned} R_x &= \dfrac{nx_1 + mx_2}{m+n} \\[0.6em] &= \dfrac{1(-2) + 3(2)}{3+1} \\[0.6em] &= \dfrac{-2 + 6}{4} \\[0.6em] &= \dfrac{4}{4} \\[0.6em] &= 1 \end{aligned}

\begin{aligned} R_y &= \dfrac{ny_1 + my_2}{m+n} \\[0.6em] &= \dfrac{1(2) + 3(8)}{3+1} \\[0.6em] &= \dfrac{2 + 24}{4} \\[0.6em] &= \dfrac{26}{4} \\[0.6em] &= \dfrac{13}{2} \end{aligned}

Thus, $R = \left(1, \dfrac{13}{2}\right)$ .

Answer

$\left(-1, \dfrac{7}{2}\right)$ , $(0, 5)$ , and $\left(1, \dfrac{13}{2}\right)$

Common Mistakes

Ratio Confusion: Dividing a line into $4$ equal parts requires ratios of $1:3$ , $2:2$ , and $3:1$ , not $1:4$ or $1:2$ .
Alternative Midpoint Method: You can also find $Q$ as the midpoint of $AB$ , then $P$ as the midpoint of $AQ$ , and $R$ as the midpoint of $QB$ to simplify calculations.

More questions in Exercise 7.2

Find the coordinates of the point which divides the join of $(-1, 7)$ and $(4, -3)$ in the ratio $2 : 3$ .

Find the coordinates of the points of trisection of the line segment joining $(4, -1)$ and $(-2, -3)$ .

To conduct Sports Day activities, in your rectangular shaped school ground $ABCD$ , lines have been drawn with chalk powder at a distance of $1\text{ m}$ each. $100$ flower pots have been placed at a distance of $1\text{ m}$ from each other along $AD$ , as shown in Fig. 7.12. Niharika runs $\dfrac{1}{4}\text{th}$ the distance $AD$ on the 2nd line and posts a green flag. Preet runs $\dfrac{1}{5}\text{th}$ the distance $AD$ on the eighth line and posts a red flag. What is the distance between both the flags? If Rashmi has to post a blue flag exactly halfway between the line segment joining the two flags, where should she post her flag?

Find the ratio in which the line segment joining the points $(-3, 10)$ and $(6, -8)$ is divided by $(-1, 6)$ .

Find the ratio in which the line segment joining $A(1, -5)$ and $B(-4, 5)$ is divided by the $x$ -axis. Also find the coordinates of the point of division.

If $(1, 2)$ , $(4, y)$ , $(x, 6)$ and $(3, 5)$ are the vertices of a parallelogram taken in order, find $x$ and $y$ .

Find the coordinates of a point $\text{A}$ , where $\text{AB}$ is the diameter of a circle whose centre is $(2, -3)$ and $\text{B}$ is $(1, 4)$ .

If A and B are $(-2, -2)$ and $(2, -4)$ , respectively, find the coordinates of P such that $\text{AP} = \dfrac{3}{7} \text{AB}$ and P lies on the line segment AB.

Find the coordinates of the points which divide the line segment joining $A(-2, 2)$ and $B(2, 8)$ into four equal parts.

Q10

Find the area of a rhombus if its vertices are $(3, 0)$ , $(4, 5)$ , $(-1, 4)$ and $(-2, -1)$ taken in order.

[Hint : Area of a rhombus = $\dfrac{1}{2}$ (product of its diagonals)]