Parallel and perpendicular lines
Two non-vertical lines are parallel when their slopes are equal (m₁ = m₂). They are perpendicular when m₁ × m₂ = −1.
Parallel or perpendicular?
These rules are for non-vertical lines. A vertical line has no slope, so check it on its own: it is perpendicular to every horizontal line.
Proving shapes with slopes
| Shape | What to show with slopes |
|---|---|
| Parallelogram | Both pairs of opposite sides are parallel |
| Rectangle | A parallelogram with two adjacent sides perpendicular |
| Rhombus | A parallelogram whose diagonals are perpendicular |
| Square | A rectangle whose diagonals are perpendicular |
| Right triangle | Two of its sides are perpendicular |
| Trapezium | One pair of opposite sides is parallel and the other pair is not |
Example 1: find k for perpendicular lines
The line through A(−1, 3) and B(3, 5) is perpendicular to the line through C(2, 7) and D(k, −1). Find k.
- Formula: m = (y₂ − y₁)/(x₂ − x₁). Perpendicular lines: m₁ × m₂ = −1
- Slope of AB = (5 − 3)/(3 − (−1)) = 2/4 = 1/2
- Slope of CD = (−1 − 7)/(k − 2) = −8/(k − 2)
- Perpendicular lines: (1/2) × (−8/(k − 2)) = −1
- So −8/(k − 2) = −2
- k − 2 = 4, so k = 6
k = 6
Example 2: is the kolam a square?
Meena draws a kolam on a square grid. Its corners are P(−2, 1), Q(2, −1), R(4, 3) and S(0, 5). Using slopes, show that PQRS is a square.
- Formula: m = (y₂ − y₁)/(x₂ − x₁). Parallel lines have equal slopes; perpendicular lines have m₁ × m₂ = −1.
- Slope PQ = (−1 − 1)/(2 − (−2)) = −2/4 = −1/2
- Slope QR = (3 − (−1))/(4 − 2) = 4/2 = 2
- Slope RS = (5 − 3)/(0 − 4) = 2/(−4) = −1/2
- Slope SP = (1 − 5)/(−2 − 0) = −4/(−2) = 2
- PQ ∥ RS and QR ∥ SP, so PQRS is a parallelogram.
- Slope PQ × slope QR = (−1/2) × 2 = −1, so PQ ⊥ QR. PQRS is a rectangle.
- Diagonals: slope PR = (3 − 1)/(4 − (−2)) = 2/6 = 1/3 and slope QS = (5 − (−1))/(0 − 2) = 6/(−2) = −3.
- Slope PR × slope QS = (1/3) × (−3) = −1, so the diagonals are perpendicular.
PQRS is a rectangle with perpendicular diagonals, so it is a square.
Railway tracks are parallel lines: same slope, never meet. The lines of a kolam grid and the corners of a room meet at right angles, so they are perpendicular.
2 marks: “Show that the line through (3, −2) and (−1, 4) is parallel to the line through (−2, 3) and (2, −3).” m₁ = (4 + 2)/(−1 − 3) = 6/(−4) = −3/2. m₂ = (−3 − 3)/(2 + 2) = −6/4 = −3/2. Since m₁ = m₂, the lines are **parallel**.
2 marks: “Using slopes, show that A(1, 1), B(5, 3) and C(3, 7) form a right-angled triangle.” Slope AB = 2/4 = 1/2 and slope BC = 4/(−2) = −2. (1/2) × (−2) = −1, so AB ⊥ BC. The triangle is **right-angled at B**.
5 marks: “Using slopes, show that A(5, 3), B(0, 8), C(−1, 1) and D(4, −4) are the vertices of a rhombus.” Slopes AB = −1, CD = −1, BC = 7, DA = 7, so ABCD is a parallelogram. Diagonals: slope AC = 1/3 and slope BD = −3, product −1, so the diagonals are perpendicular. Hence ABCD is a **rhombus**.
Parallel: same slope. Perpendicular: flip it and change the sign (2/3 and −3/2).