Question

The line joining P(–4, 5) and Q(3, 2) intersects the y-axis at point R. PM and QN are perpendicular from P and Q on the x-axis Find:

1. the ratio PR : RQ
2. the coordinates of R.
3. the area of the quadrilateral PMNQ.

Answer 1

i. Let point R (0, y) divides PQ in the ratio k : 1.

We have:

`x = (k xx 3 + 1 xx (-4))/(k + 1)`

`0 = (3k - 4)/(k + 1)`

`0 = 3k - 4`

`k = 4/3`

Thus, PR : RQ = 4 : 3

ii. Also, we have:

`y = (k xx 2 + 1 xx 5)/(k + 1)`

`y = (2k + 5)/(k + 1)`

`y = (2 xx 4/3 + 5)/(4/3 + 1)`

`y = (8 + 15)/(4 + 3)`

`y = 23/7`

Thus, the co-ordinates of point R are `(0, 23/7)`

iii. Area of quadrilateral PMNQ

= `1/2 xx (PM + QN) xx MN`

= `1/2 xx (5 + 2) xx 7`

= `1/2 xx 7 xx 7`

= 24.5 sq. units