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 coordinates of vertices of the triangle, the middle points of whose sides are

(0, \frac{1}{2}), (\frac{1}{2}, \frac{1}{2})

and

(\frac{1}{2}, 0)

.

Accepted Answer

Given, coordinates of midpoints of $∆ ABC$ are $(0, \frac{1}{2}), (\frac{1}{2}, \frac{1}{2})$ and $(\frac{1}{2}, 0)$ of sides $AB, BC$ and $CA$ as shown below.

Let the vertices be $A (x_{1}, y_{1}), B (x_{2}, y_{2}), C (x_{3}, y_{3})$ , therefore

Since $(\frac{1}{2}, \frac{1}{2})$ is the midpoint of $AB$ so using midpoint theorem we get:

$\frac{x + x_{1}}{2} = \frac{1}{2}$ and $\frac{y + y_{1}}{2} = \frac{1}{2}$

$x + x_{1} = 1$ _____(i) and $(\frac{1}{2}, 0)$ _____(ii)

Since $(0, \frac{1}{2})$ is the midpoint of $AC$ so using midpoint theorem we get:

$\frac{x + x_{2}}{2} = 0$ and $\frac{y + y_{2}}{2} = \frac{1}{2}$

$x + x_{2} = 0$ _____(iii) and $y + y_{2} = 1$ _____(iv)

Since $(\frac{1}{2}, 0)$ is the midpoint of $BC$ so using midpoint theorem we get:

$\frac{x_{1} + x_{2}}{2} = \frac{1}{2}$ and $\frac{y_{1} + y_{2}}{2} = 0$

$x_{1} + x_{2} = 1$ _____(v) and $y_{1} + y_{2} = 0$ _____(vi)

On adding equation (i), (iii) and (v) we get $x + x_{1} + x_{2} = 1$ ______(vii)

On subtracting one by one equation (i), (iii) and (v) from (vii) we get $x_{2} = 0$ , $x_{1} = 1$ and $x = 0$

On adding equation $(2), (4)$ and $(6)$ we get $y + y_{1} + y_{2} = 2$ _____(viii)

On subtracting one by one (ii), (iv) and (vi) from (viii) we get $y_{2} = 0, y_{1} = 0, y = 1$

Hence, the coordinates of the vertices of the triangle are $(0, 1), (1, 0), (0, 0)$ .

Find coordinates of vertices of the triangle, the middle points of whose sides are $(0, \frac{1}{2}), (\frac{1}{2}, \frac{1}{2})$ and $(\frac{1}{2}, 0)$ .

Important Questions on Section and Mid-Point Formulae

Learn and Prepare for any exam you want !

Find coordinates of vertices of the triangle, the middle points of whose sides are 0,12,12,12 and 12,0.

Important Questions on Section and Mid-Point Formulae

Learn and Prepare for any exam you want !

Find coordinates of vertices of the triangle, the middle points of whose sides are $(0, \frac{1}{2}), (\frac{1}{2}, \frac{1}{2})$ and $(\frac{1}{2}, 0)$ .