Question

If points A (5, 2), B (3, 4) and C (x, y) are collinear and AB = BC then find (x, y).

Answer 1

Hint: Three or more points are said to be collinear if they lie on a straight line.If three points are collinear it means the area formed by the line joining three points is zero, and as AB=BC you would get an equation. And by using this equation we can find the value of x and y.

Complete step-by-step answer:

In this figure, A, B, C are collinear points .
Given, the points A (5,2), B (3,4) and C (x, y) are collinear i.e., A, B, and C are on same line and AB=BC
∴ B is the midpoint of AC.
We know that according to Midpoint formula:
If the coordinates of A and B are ${\text{(}}{{\text{x}}_{\text{1}}}{\text{, }}{{\text{y}}_{\text{1}}}{\text{)}}$ and ${\text{(}}{{\text{x}}_{\text{2}}}{\text{, }}{{\text{y}}_{\text{2}}}{\text{)}}$respectively, then the midpoint M of AB is given by the following formula (Midpoint Formula).
${\text{M = (}}\dfrac{{{{\text{x}}_{\text{1}}}{\text{ + }}{{\text{x}}_{\text{2}}}}}{{\text{2}}}{\text{,}}\dfrac{{{{\text{y}}_{\text{1}}}{\text{ + }}{{\text{y}}_{\text{2}}}}}{{\text{2}}}{\text{)}}$
So we can write:
    ${\text{(}}\dfrac{{{\text{x + 5}}}}{{\text{2}}}{\text{,}}\dfrac{{{\text{y + 5}}}}{{\text{2}}}{\text{) = (3,4)}}$
On equating the coordinates on both sides, we get:
$ \Rightarrow $$\dfrac{{{\text{x + 5}}}}{{\text{2}}}{\text{ = 3 and }}\dfrac{{{\text{y + 2}}}}{{\text{2}}}{\text{ = 4}}$
 $ \Rightarrow $${\text{x + 5 = 6 and y + 2 = 8}}$
 $ \Rightarrow $${\text{x = 1 and y = 6}}$
Hence, the value of x and y are 1 and 6 respectively.

Note: In this question three points are collinear so the section formula will give the answer. So you should remember the sectional formula to solve this type of question. Midpoint formula itself comes from the sectional formula if the ratio is 1:1. In case these three points are non-collinear, then we will get a closed bounded by three line segments . This closed figure is called a triangle. In such a case, you have to use a formula for finding the area of the triangle to solve the problem.