Question

In triangle ABC, the midpoints of the sides AB, BC and AC are (1,4) , (2,0) and (-4,1) , respectively. Find the coordinates of points A, B and C.

Answer 1

The coordinates are A(-5,3), B(7,1) and C(-3,-1).

Explanation:

Given,

The midpoints of the sides AB, BC and AC are (1,4) , (2,0) and (-4,1) respectively.

To find the coordinates of A, B, C.

Let,

The coordinates of A,B and C are (a,x), (b,y) and (c,z) respectively.

Formula

The mid point of two given points (
`x_(1) ,y_(1)`) and (
`x_(2) ,y_(2)`) is (
`(x_(1) +x_(2) )/(2), (y_(1) +y_(2) )/(2)`)

Now,

`(a+b)/(2)` = 1 and
`(x+y)/(2)` = 4 ⇒ a+b = 2 ---(1) and x+y = 4 --(2)

`(b+c)/(2)` = 2 and
`(y+z)/(2)` = 0 ⇒ b+c = 4 ---(3) and y+z = 0---(4)

`(c+a)/(2)` = -4 and
`(z+x)/(2)` = 1 ⇒ c+a = -8 ---(5) and z+x = 2 ----(6)

Adding (1), (3) and (5) we get,

2(a+b+c) = -2

or, a+b+c = -1

So, a = -5, b = 7 and c = -3

Similarly Adding (2), (4) and (6) we get,

2(x+y+z) = 6

or, x+y+z = 3

So, x = 3, y = 1 and z = -1

Hence,

The coordinates are A(-5,3), B(7,1) and C(-3,-1).