Coordinate Geometry | Exercise 7.2

Question 2

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

Solution

Understand the Question

Trisection means dividing a line segment into three equal parts ( $AP = PQ = QB$ ).
Two points, $P$ $P$ and $Q$ $Q$ , are required to trisect a segment $AB$ $A B$ :
- Point $P$ divides $AB$ internally in the ratio $1 : 2$ .
- Point $Q$ divides $AB$ internally in the ratio $2 : 1$ (or acts as the midpoint of $PB$ ).
We use the Section Formula for a point $(x, y)$ dividing $A(x_1, y_1)$ and $B(x_2, y_2)$ in ratio $m : n$ : $x = \dfrac{mx_2 + nx_1}{m + n}, \quad y = \dfrac{my_2 + ny_1}{m + n}$

Step 1 · Find Coordinates of the First Point of Trisection $P$

Let the given points be $A(4, -1)$ and $B(-2, -3)$ .Point $P$ divides $AB$ internally in the ratio $1 : 2$ . Here, $(x_1, y_1) = (4, -1)$ , $(x_2, y_2) = (-2, -3)$ , $m = 1$ , and $n = 2$ .

Using the section formula:

\begin{aligned} x &= \dfrac{1(-2) + 2(4)}{1 + 2} \\[0.6em] &= \dfrac{-2 + 8}{3} \\[0.6em] &= \dfrac{6}{3} \\[0.6em] &= 2 \end{aligned}

\begin{aligned} y &= \dfrac{1(-3) + 2(-1)}{1 + 2} \\[0.6em] &= \dfrac{-3 - 2}{3} \\[0.6em] &= -\dfrac{5}{3} \end{aligned}

Therefore, the coordinates of $P$ are $\left(2, -\dfrac{5}{3}\right)$ .

Step 2 · Find Coordinates of the Second Point of Trisection $Q$

Point $Q$ divides $AB$ internally in the ratio $2 : 1$ . Here, $(x_1, y_1) = (4, -1)$ , $(x_2, y_2) = (-2, -3)$ , $m = 2$ , and $n = 1$ .

Using the section formula:

\begin{aligned} x &= \dfrac{2(-2) + 1(4)}{2 + 1} \\[0.6em] &= \dfrac{-4 + 4}{3} \\[0.6em] &= \dfrac{0}{3} \\[0.6em] &= 0 \end{aligned}

\begin{aligned} y &= \dfrac{2(-3) + 1(-1)}{2 + 1} \\[0.6em] &= \dfrac{-6 - 1}{3} \\[0.6em] &= -\dfrac{7}{3} \end{aligned}

Therefore, the coordinates of $Q$ are $\left(0, -\dfrac{7}{3}\right)$ .

Answer

$\left(2, -\dfrac{5}{3}\right) \text{ and } \left(0, -\dfrac{7}{3}\right)$

Common Mistakes

Ratio Confusion: Using a $1 : 3$ ratio instead of $1 : 2$ and $2 : 1$ . Trisection divides the line into $3$ equal parts, making the internal section ratios $1 : 2$ and $2 : 1$ .
Sign Errors: Mishandling negative signs when substituting coordinates into $m x_2 + n x_1$ (e.g., writing $1(-3) + 2(-1)$ as $-3 + 2$ instead of $-3 - 2$ ).
Swapping $x_1$ and $x_2$ : Multiplying $m$ with $x_1$ instead of $x_2$ in the formula $\dfrac{mx_2 + nx_1}{m + n}$ .

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)]