Question

The origin o (0, O), P (-6, 9) and Q (12, -3) are vertices of triangle OPQ. Point M divides OP in the ratio 1: 2 and point N divides OQ in the ratio 1: 2. Find the coordinates of points M and N. Also, show that 3MN = PQ. 

Answer 1

It is given that M divides OP in the ratio 1: 2 and point N divides OQ in the ratio 1: 2.

Using section formula, the coordinates of M are 

`((-6 + 0)/ 3 , (9 + 0)/3) = (-2 , 3)`

Using section formula, the coordinates of N are 

`((12 + 0)/3 , (-3 + 0)/3) = (4 , -1)`

Thus, the ooordinates of M and N are ( -2, 3) and ( 4, -1) respectively. 

Now, using distance formula, we have: 

PQ = `sqrt ((-6 -12)^2 + (9 + 3)^2) = sqrt (324 + 144) = sqrt 468`

MN = `sqrt ((4 + 2)^2 + (-1-3)^2) = sqrt (36 + 36) = sqrt 52`

It can be observed that : 

PQ = `sqrt 468 = sqrt (9 xx 52) = 3 sqrt 52 = 3  "MN"`

Hence proved.