Question

Points P(6, 0), Q(2, 8) and R(–2, 4) are vertices of ΔPQR. It is given that MN || QR such that `(PM)/(MQ) = 1/3`. Using distance formula and ratio formula, show that `(MN)/(QR) = 1/4`.

Answer 1

Given that MN || OR

∴ By basic proportionality theorem

`(PM)/(MQ) = (PN)/(NR)`

⇒ `1/3 = (PN)/(NR)`

So, PN : NR = 1 : 3

P(6, 0), R(–2, 4) and PN : NR = 1 : 3

So, by section formula, coordinates of N are

`N = ((1 xx (-2) + 3 xx 6)/(1 + 3), (1 xx 4 + 3 xx 0)/(1 + 3))`

= `((-2 + 18)/4, 4/4)`

= (4, 1)

And by section formula, coordinates of M are

`M = ((1 xx (-2) + 3 xx 6)/(1 + 3), (1 xx 8 + 3 xx 0)/(1 + 3))`

= `((2 + 18)/4, 8/4)`

= (5, 2)

Now, coordinates of M and N are (5, 2) and (4, 1) respectively

So, `MN = sqrt((5 - 4)^2 + (2 - 1)^2`

⇒ `MN = sqrt(1 + 1)`

⇒ `MN = sqrt(2)`

Coordinates of Q and R are (2, 8) and (–2, 4) respectively.

So, `QR = sqrt((2 + 2)^2 + (8 - 4)^2`

⇒ `QR = sqrt(16 + 16)`

⇒ `QR = sqrt(2 xx 16)`

⇒ `QR = 4sqrt(2)`

So, `(MN)/(QR) = sqrt(2)/(4sqrt(2))`

⇒ `(MN)/(QR) = 1/4`, hence proved.