Question

in a triangle ABC a=(5,8)D is the mid point of side bc and d(2,-1) find the centroid of abc​

Answer 1

$\textbf{Concept:}$

$\text{The centroid of triangle divides each median in the ratio 2:1}$

$\textbf{Formula used:}$

$\boxed{\begin{minipage}{8cm}\textbf{The coordinates of the point which divides the line segment joining} \bf(x_1,y_1)\:\textbf{and}\bf(x_2,y_2)\textbf{internally in the ratio m:n is}\\\\\bf(\frac{mx_2+nx_1}{m+n},\frac{my_2+ny_1}{m+n})\end{minipage}}$

$\textbf{Given:}$

$\text{In \triangle ABC, A(5,8) and}$

$\text{The midpoint of BC, D(2,-1)}$

$\text{Let G be the centroid of \triangle ABC}$

$\text{Then, G divides the median AD in the ratio 2:1}$

$\implies\text{G is}\;(\frac{mx_2+nx_1}{m+n},\frac{my_2+ny_1}{m+n})$

$\implies\text{G is}\;(\frac{2(2)+1(5)}{2+1},\frac{2(-1)+(8)}{2+1})$

$\implies\text{G is}\;(\frac{9}{3},\frac{6}{3})$

$\implies\textbf{G is}\bf(3,2)$

Find more:

Find the third vertex of a triangle, if two of its vertices are at (-3, 1) and (0, -2) and the centroid is at the origin.

https://brainly.in/question/15938718

Answer 2

Step-by-step explanation:

Given that in triangle ABC, the co-ordinates of vertex A are (5, 8) and the co-ordinates of the mid-point D of the opposite side BC are (2, -1).

So, AD is a median of triangle ABC.

We know that

Centroid divides a median in the ratio 2 : 1.

Therefore, the co-ordinates of the centroid of triangle ABC are

$\left(\dfrac{2\times2+1\times5}{2+1},\dfrac{2\times(-1)+1\times8}{2+1}\right)\\\\\\=\left(\dfrac{4+5}{3},\dfrac{-2+8}{3}\right)\\\\\\=\left(\dfrac{9}{3},\dfrac{6}{3}\right)\\\\=(3,2).$

Thus, the required co-ordinates of the centroid of triangle ABC are (3, 2).