Question

The origin O (0, 0), 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 coordinates of M and N are ( −2, 3) and ( 4, −1), respectively. 

Now, using the 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 + 16)`

= `sqrt 52`

It can be observed that: 

PQ = `sqrt 468`

= `sqrt (9 xx 52)`

= `3 sqrt 52`

= 3 MN

Hence proved.