If A (4,-8) , B (-9,7) and C(18 , 13) are the vertices of triangle ABC, find the lenght of - Maths - Coordinate Geometry

Question

If A (4,-8) , B (-9,7) and C(18 , 13) are the vertices of
triangle ABC, find the lenght of the median through A and
coordinate of centroid of triangle

Aakash Sharma · Accepted Answer

Let AD be the median through A

Then D will be the mid point of BC and coordinates of D are $(\frac{- 9 + 18}{2}, \frac{7 + 13}{2}) = (\frac{9}{2}, \frac{20}{2}) = (4.5, 10)$

Thus length of median through A = AD

$= \sqrt{(4.5 - 4)^{2} + (10 - (- 8))^{2}} = \sqrt{{(.5)}^{2} + (18)^{2}} = \sqrt{.25 + 324} = \sqrt{324.25} units$

Hence the length of the median through A is approximately $\sqrt{324.25} units$

Let E be the centroid of ∆ABC. Then E lies on median AD and divide it in the ratio 2 : 1. So the coordinates of E are

$(\frac{2 \times 4.5 + 1 \times 4}{2 + 1}, \frac{2 \times 10 + 1 \times (- 8)}{2 + 1}) = (\frac{9 + 4}{3}, \frac{20 - 8}{3}) = (\frac{13}{3}, \frac{12}{3}) = (\frac{13}{3}, 4)$

If A (4,-8) , B (-9,7) and C(18 , 13) are the vertices of triangle ABC, find the lenght of the median through A and coordinate of centroid of triangle.