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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:
- the ratio PR : RQ
- the coordinates of R.
- the area of the quadrilateral PMNQ.
Solution
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
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