Grade 10 Analytical Geometry

Grade 10 analytical geometry: the distance formula, midpoint formula and gradient, plus parallel and perpendicular lines and collinear points, with examples.

Analytical geometry describes shapes on the Cartesian plane with coordinates, and it is about 15 marks of Grade 10 Paper 2. You use three formulas, distance, midpoint and gradient, to prove properties of triangles and quadrilaterals, find missing coordinates and decide whether lines are parallel or perpendicular.

Grade 11 adds the equation of a line and its angle of inclination, and Grade 12 adds circles and tangents, so these three formulas are used for the rest of high school.

Key concepts

  • Distance: d = √[(x₂ − x₁)² + (y₂ − y₁)²]
  • Midpoint: M = ((x₁ + x₂)/2; (y₁ + y₂)/2)
  • Gradient: m = (y₂ − y₁)/(x₂ − x₁)
  • Parallel lines have equal gradients
  • Perpendicular lines have gradients whose product is −1
  • Collinear points lie on one straight line

Worked examples

Find the midpoint of A(−2; 5) and B(6; −1).

M = ((−2 + 6)/2; (5 + (−1))/2) = (2; 2).

Find the length of PQ with P(1; 2) and Q(4; 6).

PQ = √[(4 − 1)² + (6 − 2)²] = √(9 + 16) = √25 = 5 units.

Are the lines through A(0; 1), B(2; 5) and C(1; 0), D(5; −2) perpendicular?

m_AB = (5 − 1)/(2 − 0) = 2. m_CD = (−2 − 0)/(5 − 1) = −1/2. Since 2 × (−1/2) = −1, AB ⊥ CD.

Show that A(1; 1), B(3; 5) and C(4; 7) are collinear.

m_AB = (5 − 1)/(3 − 1) = 2 and m_BC = (7 − 5)/(4 − 3) = 2. The gradients are equal and the lines share the point B, so A, B and C lie on one straight line.

Common mistakes

  • Mixing the order of the points in the gradient formula (top and bottom must use the same order).
  • Forgetting the square root in the distance formula.
  • Thinking perpendicular gradients are just negatives of each other (they are negative reciprocals).
  • Sign slips when subtracting negative coordinates.

Exam tips

  • Sketch the points first; a rough diagram catches sign errors.
  • To prove a parallelogram, show that the diagonals have the same midpoint.
  • Leave lengths in surd form unless the question asks for a decimal.

Frequently asked questions

How do I prove a quadrilateral is a parallelogram?

Show that both pairs of opposite sides are parallel (equal gradients), or that the diagonals bisect each other (they have the same midpoint).

Do I need the equation of a line in Grade 10?

You use y = mx + c with functions. Finding the equation of a line from two points and the angle of inclination are developed in Grade 11 analytical geometry.

How many marks is analytical geometry in Grade 10?

About 15 of the 100 marks in Paper 2.

What does collinear mean?

Points are collinear if they lie on the same straight line. You prove it by showing that the gradient between the first and second points equals the gradient between the second and third points.

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