4PM1/01 Jan 2023, Question 2: Perpendicular Lines
Pearson Edexcel International GCSE Further Pure Mathematics 4PM1 · Paper 1 · Calculator · 13 marks
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The question, in short
Given points A(−5, 3), B(4, 0), C(−1, 5), find the line through C perpendicular to AB, its intersection with AB, check bisector status, and find tan of angle ABC.
Summarised in our own words. Use the official paper for the exact wording and any diagrams.
Part (a) · 4 marks
Find an equation of the line l through C perpendicular to AB, giving it as ax + by + c = 0 with integer coefficients.
- 1Gradient of AB = (0 − 3)/(4 + 5) = −1/3
- 2Gradient of l = 3
- 3y − 5 = 3(x + 1)
- 43x − y + 8 = 0
Answer3x − y + 8 = 0
Part (b) · 3 marks
Show that the intersection point D of line l with AB has coordinates (−2, 2).
- 1AB: y = −(1/3)(x − 4)
- 2l: y = 3x + 8
- 33x + 8 = −x/3 + 4/3, so 9x + 24 = −x + 4
- 4x = −2, y = 3(−2) + 8 = 2, so D = (−2, 2)
AnswerD = (−2, 2) shown
Part (c) · 2 marks
Show that line l is not the perpendicular bisector of segment AB.
- 1Midpoint of AB = ((−5 + 4)/2, (3 + 0)/2) = (−1/2, 3/2)
- 2This is not D (−2, 2), so l does not pass through the midpoint of AB
AnswerMidpoint of AB is (−1/2, 3/2) ≠ D, so l is not the perpendicular bisector
Part (d) · 4 marks
Find tan ∠ABC, giving the answer in its simplest form.
- 1Triangle BDC has a right angle at D
- 2CD = √((−1 + 2)² + (5 − 2)²) = √10
- 3BD = √((4 + 2)² + (0 − 2)²) = √40 = 2√10
- 4tan ∠ABC = CD/BD = √10/(2√10) = 1/2
Answertan ∠ABC = 1/2
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