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Grade 10 · Analytical Geometry · 15 min read

Analytical Geometry: All the Exam Question Types

The three core formulas — distance, midpoint, gradient — and every way an exam uses them: parallel and perpendicular lines, collinear points, and finding an unknown coordinate, each worked in full with a worksheet.

Analytical (coordinate) geometry works out geometric facts using algebra, by placing points on the Cartesian plane. Almost every question uses one of three formulas, applied to two points (x1, y1) and (x2, y2). This guide covers the formulas and every question type built on them.

1The three formulas

Distance d = (x2 − x1)2 + (y2 − y1)2
Midpoint M = ( x1 + x22 , y1 + y22 )  •  Gradient m = y2 − y1x2 − x1
xyABΔxΔy
Distance uses the horizontal gap Δx and vertical gap Δy (Pythagoras); gradient is Δy ÷ Δx.

2Type 1: Distance, midpoint and gradient of two points

Worked ExampleExample 1: A(1, 2) and B(4, 6)
  1. 1
    Distance: d = (4−1)2 + (6−2)2 = 9 + 16 = 25 = 5.
    Pythagoras on the horizontal gap (3) and vertical gap (4).
  2. 2
    Midpoint: ((1+4)/2 , (2+6)/2) = (2,5, 4).
    Average each coordinate.
  3. 3
    Gradient: m = (6−2)/(4−1) = 43.
    Rise over run — keep the points in the same order.

3Type 2: Parallel lines (equal gradients)

Two lines are parallel when their gradients are equal.

Worked ExampleExample 2: is the line through A(0, 1), B(2, 5) parallel to the one through C(1, 0), D(3, 4)?
  1. 1
    Gradient of AB: (5−1)/(2−0) = 42 = 2.
    Find each gradient.
  2. 2
    Gradient of CD: (4−0)/(3−1) = 42 = 2.
  3. 3
    Equal gradients (both 2) → the lines are parallel.

4Type 3: Perpendicular lines (gradients multiply to −1)

Two lines are perpendicular when their gradients multiply to −1 (each is the negative reciprocal of the other).

Worked ExampleExample 3: a line has gradient 3. What gradient is perpendicular to it?
  1. 1
    Perpendicular gradient × 3 = −1.
    Product of perpendicular gradients is −1.
  2. 2
    So the perpendicular gradient = −13.
    The negative reciprocal of 3.

5Type 4: Collinear points (on the same straight line)

Worked ExampleExample 4: are A(1, 2), B(3, 6), C(5, 10) collinear?
  1. 1
    Gradient AB: (6−2)/(3−1) = 42 = 2.
    Points are collinear if the gradients between them are equal.
  2. 2
    Gradient BC: (10−6)/(5−3) = 42 = 2.
  3. 3
    Equal gradients → the three points are collinear.
    They lie on one straight line.

6Type 5: Find an unknown coordinate

Worked ExampleExample 5: the midpoint of A(2, 3) and B(x, 9) is (5, 6). Find x
  1. 1
    The midpoint x-value: (2 + x)/2 = 5.
    Use the midpoint formula on the x-coordinates.
  2. 2
    2 + x = 10 → x = 8.
    Solve the equation.
Worked ExampleExample 6: the distance from (0, 0) to (x, 4) is 5. Find x (x > 0)
  1. 1
    (x−0)2 + (4−0)2 = 5 → x2 + 16 = 25.
    Square both sides to remove the root.
  2. 2
    x2 = 9 → x = 3.
    Take the positive root (given x > 0).
💡

Parallel → equal gradients; perpendicular → gradients multiply to −1. These two facts unlock most 'prove it' questions about shapes (e.g. showing a quadrilateral is a rectangle).

⚠️

Keep the points in the same order on the top and bottom of the gradient fraction. Swapping only x or only y flips the sign and gives the wrong gradient.

Analytical geometry expands to lines, circles and tangents in Grades 11–12, all built on these three formulas. Master every type below.

Practice exercises

Work each one out, then click to reveal the answer.

📄 Worksheet✅ Memo📝 Exam paper🗝️ Marking guide
  1. 1
    Distance between (1, 1) and (4, 5). (distance)
    Show answer ▾
    9+16 = 25 = 5
  2. 2
    Distance between (−2, 1) and (1, 5). (distance)
    Show answer ▾
    9+16 = 5
  3. 3
    Midpoint of (2, 4) and (6, 10). (midpoint)
    Show answer ▾
    ((2+6)/2,(4+10)/2) = (4, 7)
  4. 4
    Gradient of the line through (1, 2) and (5, 10). (gradient)
    Show answer ▾
    (10−2)/(5−1) = 2
  5. 5
    Are gradients 3 and 3 parallel or perpendicular? (parallel)
    Show answer ▾
    Equal → parallel
  6. 6
    What gradient is perpendicular to gradient 2? (perpendicular)
    Show answer ▾
    −12 (negative reciprocal)
  7. 7
    What gradient is perpendicular to gradient −34? (perpendicular)
    Show answer ▾
    4⁄3
  8. 8
    Are (0, 0), (2, 4), (3, 6) collinear? (collinear)
    Show answer ▾
    Gradients 2 and 2 → Yes
  9. 9
    Midpoint of (1, 2) and (x, 8) is (4, 5). Find x. (unknown)
    Show answer ▾
    (1+x)/2 = 4 → x = 7
  10. 10
    Distance from (0, 0) to (x, 12) is 13 (x > 0). Find x. (unknown)
    Show answer ▾
    x2 + 144 = 169 → x = 5

Now practise it

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