Name: Coordinates of Centroid
Uploaded: 2019-07-19
Duration: 3 min 16 s
Description: This video discusses how we can find the coordinates of the centroid using the coordinates of the three vertices given. Watch and learn the concept.

Question

Find the coordinates of point equidistant from the three given points

A (5, 1), B (- 3, 7)

and

C (7, - 1)

Accepted Answer

Let the co-ordinates of the point equidistant from three given points $A (5, 1), B (- 3, - 7)$ and $C (7, - 1)$ be $P (x, y)$ .

From distance formula, we know that distance between two points $A (x_{1}, y_{1})$ and $B (x_{2}, y_{2})$ is

$A B = \sqrt{{(x_{2} - x_{1})}^{2} + {(y_{2} - y_{1})}^{2}}$

As per given, we get, $A P = B P = C P$ $\Rightarrow A P^{2} = B P^{2} = C P^{2}$

Consider $A (5, 1), B (- 3, 7)$
$\Rightarrow x^{2} + 25 - 10 x + y^{2} + 1 - 2 y = x^{2} + 9 + 6 x + y^{2} + 49 + 14 y = x^{2} + 49 - 14 x + y^{2} + 1 + 2 y$
$\Rightarrow x^{2} + 25 - 10 x + y^{2} + 1 - 2 y = x^{2} + 9 + 6 x + y^{2} + 49 + 14 y$
$\Rightarrow 25 - 10 x + 1 - 2 y = 9 + 6 x + 49 + 14 y$
$\Rightarrow 26 - 58 = 10 x + 2 y + 6 x + 14 y$
$\Rightarrow 16 x + 16 y = - 32$
$\Rightarrow x + y = - 2$
$\Rightarrow x = - 2 - y$ .......(i)

Now considering $A P^{2} = C P^{2}$
$\Rightarrow {(x - 5)}^{2} + {(y - 1)}^{2} = {(x - 7)}^{2} + {(y + 1)}^{2}$
$\Rightarrow x^{2} + 25 - 10 x + y^{2} + 1 - 2 y = x^{2} + 49 - 14 x + y^{2} + 1 + 2 y$
$\Rightarrow 25 - 10 x + 1 - 2 y = 49 - 14 x + 1 + 2 y$
$\Rightarrow 26 + 4 x - 4 y = 50$
$\Rightarrow 4 x - 4 y = 24$
$\Rightarrow x - y = 6$ ......(ii)

Substituting eq.(i) in (ii), we get,
$\Rightarrow - 2 - y - y = 6$
$\Rightarrow - 2 y = 8$
$\Rightarrow y = - 4$

From eq.(i), we get,
$x = - 2 - (- 4)$
$\Rightarrow x = 2$

$∴ (2, - 4)$ is the required point.

Find the coordinates of point equidistant from the three given points $A (5, 1), B (- 3, 7)$ and $C (7, - 1)$

Important Questions on Coordinate Geometry

Learn and Prepare for any exam you want !

Find the coordinates of point equidistant from the three given points A5, 1, B-3, 7 and C7, -1

Important Questions on Coordinate Geometry

Learn and Prepare for any exam you want !

Find the coordinates of point equidistant from the three given points $A (5, 1), B (- 3, 7)$ and $C (7, - 1)$