Grade 11 Analytical Geometry — How to Identify the Question Type

Analytical Geometry questions give you coordinates and ask you to work out a property of the line or shape they form. Each question type uses one of four formulas — distance, midpoint, gradient, or the equation of a line — and the trigger is almost always a single word: 'distance', 'midpoint', 'gradient', or 'equation'.

Type 1: Distance between two points

Trigger words: "Calculate the length of AB", "Determine the distance between A and B"

Trigger structure: Two coordinate points are given and a LENGTH is asked for.

Do not confuse with: The midpoint (Type 2) — distance gives a single number (a length), midpoint gives a coordinate point.

Method (no numbers — just the steps)

Label the two points (x1 ; y1) and (x2 ; y2)
Substitute into the distance formula: d = √[(x2 − x1)² + (y2 − y1)²]
Simplify, leaving the answer as a surd unless told to round

Grade 11Paper 2Level 1 — Knowledge

Given: A(−2 ; 1) and B(3 ; 13) Calculate the length of AB.

Practice question — not sourced from a past paper.

Common mistake

Subtracting the coordinates in the wrong order across the two brackets (e.g. x2 − x1 in one bracket but y1 − y2 in the other) — the formula is squared so the order doesn't actually change the answer, but mixing up which point is which often causes a sign error to creep in elsewhere in the same question.

See the progression — same type, increasing difficulty

Easy

Given: P(0 ; 0) and Q(6 ; 8) Calculate the length of PQ.

Practice question — not sourced from a past paper.

Medium

Given: A(1 ; 2), B(4 ; 6) and C(7 ; 2). Determine whether triangle ABC is isosceles.

Practice question — not sourced from a past paper.

Hard

Given: A(−1 ; 2) and B(k ; 10), where k > 0. The length of AB is 10 units. Calculate the value of k.

Practice question — not sourced from a past paper.

Type 2: Midpoint of a line segment

Trigger words: "Calculate the coordinates of the midpoint of AB", "M is the midpoint of AB"

Trigger structure: Two coordinate points are given and a POINT (not a length) is asked for.

Method (no numbers — just the steps)

Label the two points (x1 ; y1) and (x2 ; y2)
Substitute into the midpoint formula: M = ((x1 + x2)/2 ; (y1 + y2)/2)
Write the answer as a coordinate pair

Grade 11Paper 2Level 1 — Knowledge

Given: A(−4 ; 3) and B(2 ; −7) Calculate the coordinates of the midpoint of AB.

Practice question — not sourced from a past paper.

Common mistake

Reporting only one coordinate of the midpoint instead of the full (x ; y) pair.

See the progression — same type, increasing difficulty

Easy

Given: C(0 ; 4) and D(8 ; 0) Calculate the coordinates of the midpoint of CD.

Practice question — not sourced from a past paper.

Medium

M(3 ; 5) is the midpoint of PQ, where P(−1 ; 2). Calculate the coordinates of Q.

Practice question — not sourced from a past paper.

Hard

A(2 ; 1), B(8 ; 5) and C(4 ; k) are positioned so that C lies on the line through the midpoint of AB and is twice as far from that midpoint as A is. Calculate the value of k.

Practice question — not sourced from a past paper.

Type 3: Gradient and collinear points

Trigger words: "Calculate the gradient of AB", "Show that A, B and C are collinear"

Trigger structure: Two points are given and a gradient (steepness, or the angle a line makes with the x-axis) is asked for — or three points are given and you must show they lie on the same straight line.

Do not confuse with: The equation of a line (Type 4) — gradient is a single number; collinearity is a yes/no conclusion, not a full equation.

Method (no numbers — just the steps)

Substitute into the gradient formula: m = (y2 − y1)/(x2 − x1)
For collinear points, calculate the gradient between two different pairs of the three points
If both gradients are equal, the points are collinear (they lie on the same line)

Grade 11Paper 2Level 2 — Routine Procedures

Given: A(1 ; 1), B(3 ; 5) and C(5 ; 9) Show that A, B and C are collinear.

Practice question — not sourced from a past paper.

Common mistake

Mixing the x- and y-coordinates from different points when substituting (e.g. using x from point A but y from point C).

See the progression — same type, increasing difficulty

Easy

Given: A(2 ; 3) and B(6 ; 11) Calculate the gradient of AB.

Practice question — not sourced from a past paper.

Medium

Given: A(−2 ; 4), B(1 ; k) and C(4 ; 10). A, B and C are collinear. Calculate the value of k.

Practice question — not sourced from a past paper.

Hard

Given: A(1 ; 2), B(5 ; 4) and C(3 ; 8). Calculate the size of angle ABC, correct to ONE decimal place.

Practice question — not sourced from a past paper.

Type 4: Equation of a line

Trigger words: "Determine the equation of the line", "in the form y = mx + c"

Trigger structure: Enough information is given to fix both the gradient and a point on the line — either directly, or via parallel/perpendicular conditions.

Do not confuse with: Just calculating the gradient (Type 3) — this type asks for the FULL equation, not a single number.

Method (no numbers — just the steps)

Find the gradient, m (directly, from two points, or from a parallel/perpendicular condition)
Substitute m and one known point into y − y1 = m(x − x1)
Rearrange into the form y = mx + c

Grade 11Paper 2Level 2 — Routine Procedures

Determine the equation of the line passing through A(2 ; −1) and B(6 ; 7), in the form y = mx + c.

Practice question — not sourced from a past paper.

Common mistake

For perpendicular lines, using the same gradient as the given line instead of the negative reciprocal (m1 × m2 = −1).

See the progression — same type, increasing difficulty

Easy

Determine the equation of the line with gradient 3, passing through the point (1 ; 5).

Practice question — not sourced from a past paper.

Medium

Determine the equation of the line passing through (4 ; 2) that is PARALLEL to the line y = 2x − 3.

Practice question — not sourced from a past paper.

Hard

Line AB passes through A(−1 ; 3) and B(5 ; 6). Determine the equation of the line that passes through B and is PERPENDICULAR to AB.

Practice question — not sourced from a past paper.

Words like determine and hence appear throughout this topic — see the instruction word glossary for full definitions.