Question

A(1, 4), B(3, 2) and C(7, 5) are the vertices of a $$\triangle \mathrm{ABC}$$. 

Find:

1. the co-ordinates of the centroid G of $$\triangle \mathrm{ABC}$$, 
2. the equation of a line through G and parallel to AB.

Answer 1

i. Centroid $$\mathrm{G}(x, y)$$ of $$\triangle \mathrm{ABC}$$ is:

$$x = \frac{1 + 3 + 7}{3} = \frac{11}{3}$$ 

$$y = \frac{4 + 2 + 5}{3} = \frac{11}{3}$$

So, $$\mathrm{G} = \left(\frac{11}{3}, \frac{11}{3}\right)$$.

ii. Slope of line $$\mathrm{AB}$$: 

$$m = \frac{2 - 4}{3 - 1}$$

$$= \frac{-2}{2}$$

$$= -1$$

Since the required line is parallel to $$\mathrm{AB}$$, its slope is also $$m = -1$$. 

Passing through $$\mathrm{G}\left(\frac{11}{3}, \frac{11}{3}\right)$$: 

$$y - \frac{11}{3} = -1\left(x - \frac{11}{3}\right)$$

$$y - \frac{11}{3} = -x + \frac{11}{3}$$

$$x + y = \frac{22}{3}$$

$$3x + 3y - 22 = 0$$