Question

The points A(5,2) , B(3,4) and C(x,y) are collinear points and AB=BC,then find the co-ordinates of C.
A.(1,6)
B(2,-2)
C(4,3)
D(8,6)

Answer 1

Hint: Since the points are collinear hence all three points lie on the same line and given that AB=BC shows that B is the midpoint of AC. Use collinearity of three lines and midpoint theorem to solve this question.

Complete step-by-step answer:
Step 1:
It is given that the points A , B ,C are collinear.Three points are said to be collinear when they lie on the same line .
Therefore, A, B, C lie on the same line .
Step 2:
It is also given that AB = BC.This clearly means that the line AC is bisected at B.
Therefore, B is the midpoint of AC
Step 3:
Midpoint of the line joining two points $({x_1},{y_1}){\text{ and }}({x_2},{y_2}) = \left( {\dfrac{{{x_1} + {x_2}}}{2},\dfrac{{{y_1} + {y_2}}}{2}} \right)$
Here we have A (5,2) and c (x,y)
So the midpoint of AC is given by
$ \Rightarrow \left( {\dfrac{{{x_1} + {x_2}}}{2},\dfrac{{{y_1} + {y_2}}}{2}} \right) = \left( {\dfrac{{5 + x}}{2},\dfrac{{2 + y}}{2}} \right)$
Step 4:
As we already said B is the midpoint of AC
It is given that=(3,4)
$
   \Rightarrow B = \left( {\dfrac{{5 + x}}{2},\dfrac{{2 + y}}{2}} \right) \\
   \Rightarrow \left( {3,4} \right) = \left( {\dfrac{{5 + x}}{2},\dfrac{{2 + y}}{2}} \right) \\
    \\
$

Step 5
Now since they are equal lets equate the x and y coordinate respectively to find the values of x and y.
First let's equate the x coordinate
$ \Rightarrow 3 = \dfrac{{5 + x}}{2}$
Cross multiplying we get,
$
   \Rightarrow 6 = 5 + x \\
   \Rightarrow 6 - 5 = x \\
   \Rightarrow x = 1 \\
$
Hence the value of x is 1.
Now lets equate the y coordinate
$ \Rightarrow 4 = \dfrac{{2 + y}}{2}$
Cross multiplying we get,
$\begin{gathered}
   \Rightarrow 8 = 2 + y \\
   \Rightarrow 8 - 2 = y \\
   \Rightarrow 6 = y \\
\end{gathered} $
Hence the value of y is 6
Therefore the coordinates of C is (1,6)
The correct option is A


Note: Whenever the given three points are collinear ,then the area of the triangle formed by these points is zero since they all lie on the same line.
Many students tend to use the above property here as the points are collinear but you can use it only when one coordinate of a point is unknown, or else you will be left out with a linear equation in x and y.