4PM1/01R Jun 2024, Question 9: Length of a Line Segment

Part (a) · 2 marks

Find an equation of the line k passing through A(−4, 3) and B(6, 8).

1. 1Gradient = (8 − 3)/(6 − (−4)) = 1/2
2. 2y − 8 = ½(x − 6), i.e. y = ½x + 5

Answery − 8 = ½(x − 6) (y = ½x + 5)

Part (b) · 2 marks

Find the coordinates of C on k such that AC : CB = 4:1.

1. 1C = ((4 × 6 + 1 × (−4))/5, (4 × 8 + 1 × 3)/5)
2. 2C = (4, 7)

AnswerC(4, 7)

Part (c) · 6 marks

Find coordinates of D(p, q), p < 0, on the line through C perpendicular to k, with CD = 8√5.

1. 1Gradient of l = −2, so (q − 7)/(p − 4) = −2, q = 15 − 2p
2. 2(4 − p)² + (7 − q)² = (8√5)² = 320
3. 3(4 − p)² + (2p − 8)² = 320
4. 4p² − 8p − 48 = 0, (p + 4)(p − 12) = 0
5. 5p < 0 gives p = −4, q = 23

AnswerD(−4, 23)

Part (d) · 2 marks

Find the area of triangle ACD.

1. 1AC = √(8² + 4²) = 4√5 and angle ACD = 90°
2. 2Area = ½ × 4√5 × 8√5 = 80

Answer80 units²

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