Topic 4 of 9 · Form 3
Coordinate geometry and circles
Gradient, midpoint and distance formulae; equations y − y₁ = m(x − x₁); parallel (equal gradients) and perpendicular lines (m₁m₂ = −1); perpendicular bisectors; area of a triangle or polygon using coordinates (shoelace method).
Notes for this topic 1 file, in order
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Straight lines
Gradient, midpoint and distance formulae; equations y − y₁ = m(x − x₁); parallel (equal gradients) and perpendicular lines (m₁m₂ = −1); perpendicular bisectors; area of a triangle or polygon using coordinates (shoelace method).
Worked example: Find the perpendicular bisector of A(2, 1) and B(6, 5).
Midpoint (4, 3); gradient AB = 1 → perpendicular gradient −1 → y − 3 = −(x − 4) → y = −x + 7
Circles
(x − a)² + (y − b)² = r²; the general form x² + y² + 2gx + 2fy + c = 0 with centre (−g, −f). Tangents are perpendicular to radii; intersections of lines and circles.
Worked example: Find the centre and radius of x² + y² − 4x + 6y − 12 = 0.
(x − 2)² + (y + 3)² = 25 → centre (2, −3), radius 5
Area using coordinates
Worked example: Find the area of the triangle with vertices (0, 0), (4, 0) and (1, 3).
½|0 × 0 − 4 × 0 + 4 × 3 − 1 × 0 + 1 × 0 − 0 × 3| = ½ × 12 = 6
Check yourself
- Find the distance between (1, 2) and (7, 10).
- Find the gradient of a line perpendicular to 2x + 3y = 6.
- Write the equation of the circle centre (−1, 2) radius 4.
Show answers
- 10
- 3/2
- (x + 1)² + (y − 2)² = 16
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