Question

If A (-14, -10), B (6, -2) is given, find the coordinates of the points which divide segment AB into four equal parts?

Answer 1

We are asked to find the coordinates of the points which divide the points A and B into four equal parts. This means that three points are there. Let us name them: P, Q and R. The point which is adjacent to A is P and the point which is adjacent to B is R. Now, we are going to use the section formula to find the coordinates of P, Q and R. We know that if we have two points say, $\left( {{x}_{1}},{{y}_{1}} \right)\And \left( {{x}_{2}},{{y}_{2}} \right)$ and there is a point say $\left( a,b \right)$ which divides it into the ratio $\left( {{m}_{1}}:{{m}_{2}} \right)$ then the relation between them is as follows:
$\begin{aligned} & a=\dfrac{{{m}_{1}}{{x}_{2}}+{{m}_{2}}{{x}_{1}}}{{{m}_{1}}+{{m}_{2}}}; \\\ & b=\dfrac{{{m}_{1}}{{y}_{2}}+{{m}_{2}}{{y}_{1}}}{{{m}_{1}}+{{m}_{2}}} \\\ \end{aligned}$

Complete step by step answer:
We have given a segment AB in which coordinates A and B are given as A (-14, -10), B (6, -2). Now, we are drawing a line segment AB in the below:

Let P, Q and R divide the above line segment into four equal parts. In the above figure, we are drawing three points P, Q and R as follows:

As lengths of AP, PQ, QR and RB are equal so P divides A and B into 1:3 ratio so we are using the section formula to find the coordinates of point P. Now, let us see how the section formula is used. Let us consider two points $\left( {{x}_{1}},{{y}_{1}} \right)\And \left( {{x}_{2}},{{y}_{2}} \right)$ and there is a point say $\left( a,b \right)$ which divides it into the ratio $\left( {{m}_{1}}:{{m}_{2}} \right)$ then the relation between them is as follows:
$\begin{aligned} & a=\dfrac{{{m}_{1}}{{x}_{2}}+{{m}_{2}}{{x}_{1}}}{{{m}_{1}}+{{m}_{2}}}; \\\ & b=\dfrac{{{m}_{1}}{{y}_{2}}+{{m}_{2}}{{y}_{1}}}{{{m}_{1}}+{{m}_{2}}} \\\ \end{aligned}$
Now, let us take $\left( {{x}_{1}},{{y}_{1}} \right)\And \left( {{x}_{2}},{{y}_{2}} \right)$ as (-14, -10) and (6, -2) respectively and (a, b) are the coordinates of point P and the values of $\left( {{m}_{1}}\And {{m}_{2}} \right)$ are 1 and 3 respectively. Substituting these points in the above we get,
$\begin{aligned} & a=\dfrac{1\left( 6 \right)+3\left( -14 \right)}{1+3} \\\ & \Rightarrow a=\dfrac{6-42}{4} \\\ & \Rightarrow a=\dfrac{-36}{4}=-9 \\\ & b=\dfrac{1\left( -2 \right)+3\left( -10 \right)}{1+3} \\\ & \Rightarrow b=\dfrac{-2-30}{4} \\\ & \Rightarrow b=\dfrac{-32}{4}=-8 \\\ \end{aligned}$
From the above, we found the coordinates of P as (-9, -8).
Now, we are going to find the coordinates of Q which is the midpoint of A and B so the x-coordinates of Q is calculated by adding the x coordinates of A and B and then dividing them by 2. After that, y-coordinates is calculated by adding the y-coordinates of A and B and dividing them by 2.
$\begin{aligned} & Q=\left( \dfrac{-14+6}{2},\dfrac{-10-2}{2} \right) \\\ & \Rightarrow Q=\left( \dfrac{-8}{2},\dfrac{-12}{2} \right) \\\ & \Rightarrow Q=\left( -4,-6 \right) \\\ \end{aligned}$
From the above, we have calculated the coordinates of Q.
Now, we are going to find the coordinates of R and R is the midpoint of Q and B so the coordinates of R are as follows:
$\begin{aligned} & R=\left( \dfrac{-4+6}{2},\dfrac{-6-2}{2} \right) \\\ & \Rightarrow R=\left( \dfrac{2}{2},\dfrac{-8}{2} \right) \\\ & \Rightarrow R=\left( 1,-4 \right) \\\ \end{aligned}$
Hence, we have found the coordinates of the three points (P, Q and R) which divides the line segment AB into four equal parts.

Note: To solve the above problem, you must know what section formula is and how to use the section formula in the questions. Also, make sure you haven’t made any calculation mistakes, especially in the negative and positive signs.