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Coordinate Geometry with Line Intersections and Perpendicular Distance

Coordinate Geometry with Line Intersections and Perpendicular Distance

TGM Original

In the diagram \(R\) is the point \(\left( 0,-4 \right)\) and \(P\) is a point on the \(y\)-axis.
The line \(PQ\) meets the horizontal line through \(R\) at \(Q\).

  1. State the equation of \(QR\).
  2. Given that the equation of the line \(PQ\) is \(2y-5x-4=0\), find the coordinates of \(P\) and \(Q\).
  3. Given that the length of \(PQ\) is \(\sqrt{t}\), find the value of \(t\).
  4. Find the area of triangle \(PQR\).
  5. Hence, calculate the perpendicular distance from \(R\) to \(PQ\).

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Answer:(i) \(y = -4\) (ii) \(P(0, 2)\), \(Q\left(-\frac{12}{5}, -4\right)\) (iii) \(t = 41.76\) (iv) \(7.2\) units\(^2\) (v) \(2.23\) units

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