Question

In ABC, and intersect each other at point Q. According to a theorem on medians, Q divides in the ratio 2 : 1. What are the coordinates of Q?

Answer 1

The point of intersection of medians is a center of triangle ABC. That means that
`Q(x,y)=( (x_A+x_B+x_C)/(3) , (y_A+y_B+y_C)/(3) )`.
If A(1,-5), B(3,-2) and C(7,-5) the Q coordinates are:

`x= (1+3+7)/(3) = (11)/(3)`

`y= (-5-2-5)/(3) =-4`
Answer:
`Q( (11)/(3),-4 )` and the correct choice is D.

Answer 2

The answer is the fourth choice - (3.67, -4)

Since Q is the center of gravity of the triangle ABC, we can use this relationship - vecQA + vecQB + vecQC = vec0 : we will deduct the coordinates of A, B, and C from x and y of Q to get its coordinate.

vecQA = (x-3, y+2)
vecQB = (x-1, y+5)
vecQC = (x-7, y+5)
vec0 = (0, 0)

This will be equivalent to
(x-3) + (x-1) + (x-7) = 0, and (y+2) + (y+5) + (y+5) = 0
3x - 11 = 0, and 3y + 12 = 0
x = 11/3 or 3.67, and y = 12/3 or 4

The coordinates of Q are (3.67, 4)