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Angle between two lines is taught in the note for Coordinate Geometry. Read it below and look for the section on this topic.

Coordinate Geometry

14 min read

Angle between two lines

If two lines have gradients m₁ and m₂, the angle θ between them satisfies:

If m₁m₂ = −1 the lines are perpendicular; if m₁ = m₂ they are parallel.
Worked exampleAngle between lines

Find the angle between y = 2x and y = x/3.

  1. Gradients are 2 and 1/3.

Answer: θ = 45°.

Perpendicular distance

The shortest distance from a point (x₁, y₁) to the line ax + by + c = 0 is:

The perpendicular distance.
Worked exampleDistance from a point to a line

Find the distance from (2, 3) to the line 3x + 4y − 5 = 0.

  1. Substitute.

Answer: 2.6 units.

Locus of a moving point

A locus is the path of a point that moves according to a rule. The locus of a point at a fixed distance r from a point (a, b) is a circle: (x − a)² + (y − b)² = r². The locus of points equidistant from two points is their perpendicular bisector. The locus of points equidistant from a fixed point and a fixed line is a parabola.

Parts of a circleA circle with its parts labelled: the centre, the radius from the centre to the edge, the diameter through the centre, a chord joining two points on the circle, an arc which is part of the circumference, a sector between two radii, a segment cut off by a chord, and a tangent touching the circle at one point.centreradiusdiametersectorchordarcsegmenttangent

The ratio theorem

The point P that divides the line AB internally in the ratio m : n has coordinates:

A(x₁, y₁), B(x₂, y₂). For m = n this is the midpoint. For an external division, the ratio is taken as negative.
The locus of a point 5 units from the origin, shown as y = √(25 − x²).
Try itDistance to a line

The line 3x + 4y − 5 = 0. Choose a point.

Check yourself Lines of gradient 3 and −1/3 are
Check yourself The distance from the origin to the line 3x + 4y − 10 = 0 is
Check yourself The point dividing A(0, 0) and B(9, 6) in the ratio 1 : 2 is

Key terms

Locus
The path of a point moving by a rule.
Perpendicular distance
The shortest distance from a point to a line.
Internal division
A point between A and B in a given ratio.
Perpendicular bisector
The line perpendicular to AB through its midpoint.

Summary

  • tan θ = |(m₁ − m₂)/(1 + m₁m₂)|.
  • Distance from (x₁, y₁) to ax + by + c = 0 is |ax₁ + by₁ + c| ÷ √(a² + b²).
  • A locus is described by an equation.
  • The ratio theorem finds a dividing point.
Perpendicular distance