4PM1/01 Jun 2025, 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 A(3, 2), B(8, 3), C(4, 7), prove ABC is isosceles, find the line through A and midpoint M of BC, then find D on line AC with AD = kAC, k > 1, and angle ABD = 90.
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Part (a) · 2 marks
Show that triangle ABC with vertices A(3, 2), B(8, 3) and C(4, 7) is isosceles.
- 1AB = √((8 − 3)² + (3 − 2)²) = √26
- 2AC = √((4 − 3)² + (7 − 2)²) = √26
- 3AB = AC, so triangle ABC is isosceles
AnswerAB = AC = √26, so ABC is isosceles
Part (b) · 3 marks
M is the midpoint of BC. Find the equation of the line through A and M in the form y = mx + c.
- 1M = ((8 + 4)/2, (3 + 7)/2) = (6, 5)
- 2Gradient of AM = (5 − 2)/(6 − 3) = 1
- 3y − 2 = 1(x − 3), so y = x − 1
Answery = x − 1
Part (c) · 8 marks
A, C and D are collinear with AD = kAC (k > 1) and angle ABD = 90. Find the coordinates of D, showing working.
- 1Line AC: gradient (7 − 2)/(4 − 3) = 5, y = 5x − 13
- 2Gradient of AB = 1/5, so gradient of BD = −5
- 3Line BD: y − 3 = −5(x − 8), y = −5x + 43
- 45x − 13 = −5x + 43 gives x = 5.6
- 5y = 5(5.6) − 13 = 15, so D = (5.6, 15)
AnswerD(28/5, 15)
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