Equation of a line through two points, lines parallel and perpendicular to a given line, and the angle of inclination θ where m = tanθ.
1The Grade 10 toolkit, revised
d = (x₂ − x₁)² + (y₂ − y₁)²
m = y₂ − y₁x₂ − x₁
M = ( x₁ + x₂2 ; y₁ + y₂2 )
m = y₂ − y₁x₂ − x₁
M = ( x₁ + x₂2 ; y₁ + y₂2 )
2Equation of a line
y − y₁ = m(x − x₁)
Worked ExampleFind the equation of the line through A(−1 ; 2) and B(3 ; 10)
- 1m = 10 − 23 − (−1) = 84 = 2.Gradient first, always.
- 2y − 2 = 2(x − (−1)).Substitute either point — A is used here.
- 3y − 2 = 2x + 2.
- 4y = 2x + 4.Check with B: 2(3) + 4 = 10 ✓
3Parallel and perpendicular
Parallel: m₁ = m₂
Perpendicular: m₁ × m₂ = −1
Perpendicular: m₁ × m₂ = −1
Worked ExampleFind the line through (2 ; 5) perpendicular to y = 2x + 4
- 1m of the given line = 2.The coefficient of x.
- 2Perpendicular gradient: 2 × m = −1 → m = −12.The negative reciprocal — flip it and change the sign.
- 3y − 5 = −12(x − 2).
- 4y = −12x + 6.
4Angle of inclination
m = tanθ where 0° ≤ θ < 180°
θ is the angle the line makes with the positive x-axis, measured anticlockwise.
Worked ExampleDetermine the angle of inclination of y = 2x + 4
- 1tanθ = 2.The gradient is the tangent of the inclination.
- 2θ = tan−1(2) = 63,43°.
- 3θ ≈ 63,4°.Positive gradient ⇒ acute angle. Sensible.
Worked ExampleDetermine the angle of inclination of a line with m = −1
- 1tanθ = −1 gives ref = 45°.Take tan−1 of the positive value.
- 2A negative gradient means θ is obtuse.The line slopes down to the right.
- 3θ = 180° − 45° = 135°.
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A calculator gives tan−1(−1) = −45°. That is not an angle of inclination — θ must lie in [0° ; 180°), so add 180° to get 135°.
Practice exercises
Work each one out, then click to reveal the answer.
- 1Determine the gradient of the line through A(−1 ; 2) and B(3 ; 10).
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10 − 23 + 1 = 2 - 2Determine the equation of the line through A(−1 ; 2) and B(3 ; 10).
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y = 2x + 4 - 3Determine the length of AB where A(−1 ; 2) and B(3 ; 10).
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4² + 8² = 80 = 45 ≈ 8,94 - 4Determine the midpoint of AB where A(−1 ; 2) and B(3 ; 10).
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(1 ; 6) - 5Determine the equation of the line through (2 ; 5) perpendicular to y = 2x + 4.
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m = −12 → y = −12x + 6 - 6Determine the equation of the line through (0 ; 1) parallel to y = 2x + 4.
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y = 2x + 1 - 7Determine the angle of inclination of y = 2x + 4 (one decimal).
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tanθ = 2 → θ ≈ 63,4° - 8Determine the angle of inclination of a line with gradient −1.
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ref = 45°, obtuse → θ = 135°
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