Question

Abc is a triangle whose vertices are a(3,4), b(-2,-1) and c(5,3). If g is the centroid and bdcg is parallelogram then find the coordinates of the vertex



d.

Answer 1

Answer:

The vertices of triangle abc are a(3,4), b(-2,-1) and c(5,3).

Coordinates of Centroid(g)are

As, Centroid is the point of intersection of medians.

As, centroid divides the vertex and the point where the median intersects the other side in the ratio of 2:1.

Mid point of Segment bc is s ,

$=(\frac{-2+5}{2},\frac{-1+3}{2})=(\frac{3}{2},1)$

Given by mid point formula of line joining two points $(x_{1},y_{1}),{\text{and}},(x_{2},y_{2}) is (\frac{x_{1}+x_{2}}{2},\frac{y_{1}+y_{2}}{2}).$

So, centroid g, can be obtained by Section formula, as point a(3,4) and s$(\frac{3}{2},1)$ is given by

$[x=\frac{ma+np}{m+n}, y=\frac{mb+nq}{m+n}]\\\\g(x,y)=[\frac{\frac{3}{2}*2+1*3}{2+1},\frac{2*1+1*4}{2+1}]\\\\g(x,y)=(2,2)$  

As, it is given that , b d cg, is a parallelogram.

Let vertices of d be (f,l)

Since ,Diagonals of parallelogram bisect each other.

$\frac{f+2}{2}=\frac{5-2}{2}\\\\ f+2=3\\\\f=3-2=1\\\\ \frac{l+2}{2}=\frac{3-1}{2}\\\\l+2=2\\\\l=0$

Coordinates of vertices d of parallelogram b d cg is (1,0).