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Parallel lines have equal slopes: proofs and collinear points
Mathematics Prep 3 Term 1, Unit 3, Lesson 6
GoalUse equal slopes to show that lines are parallel and that points lie on one line.
New words
parallel lines · lines that never meet; they have the same inclinationcollinear points · points that lie on one straight lineparallelogram · a quadrilateral with both pairs of opposite sides parallelprove · show with reasons that a statement is always true
Explanation
Two non-vertical lines are parallel exactly when their slopes are equal: m₁ = m₂. Reason: parallel lines make the same angle θ with the x-axis, so tan θ is the same. To show that three points A, B, C are collinear, find the slope of AB and the slope of BC: they share the point B, so if the slopes are equal then A, B, C are on one line. To prove a quadrilateral is a parallelogram, show that both pairs of opposite sides have equal slopes. Example: A(0,0), B(4,1), C(5,4), D(1,3): slope AB = 1/4 = slope DC and slope BC = 3 = slope AD, so ABCD is a parallelogram.
Worked examples
Are the lines AB (A(1, 1), B(4, 7)) and CD (C(2, −3), D(5, 3)) parallel? Slope AB = 6/3 = 2 and slope CD = 6/3 = 2: equal, so AB ∥ CD.
Are P(−2, −3), Q(1, 3), R(4, 9) collinear? Slope PQ = 6/3 = 2 and slope QR = 6/3 = 2: yes, collinear.
Are (0, 0), (2, 3), (4, 5) collinear? Slopes 3/2 and (5 − 3)/(4 − 2) = 1 are different, so no.
Class activity
Parallel rails. Draw a grid and a line through (1, 1) with slope 2. Draw a second line through (2, −3) with the same slope. Do they ever meet?
Practice
1
Is the line through (0, 1) and (2, 5) parallel to the line through (1, 0) and (4, 6)?
Slopes 4/2 = 2 and 6/3 = 2: yes.
2
Find k so that (1, 2), (3, 6) and (5, k) are collinear.
Slope 2, so k = 6 + 2 × 2 = 10.
3
Show that (0, 0), (1, 2), (2, 4) are collinear.
Both slopes are 2.
4
Is A(0, 0), B(4, 1), C(5, 4), D(1, 3) a parallelogram? Prove it.
AB and DC have slope 1/4; BC and AD have slope 3. Yes.
5
A line has inclination 30°. What is the slope of a line parallel to it?
The same: tan 30° = 1/√3.