Grade 10 Analytical Geometry — How to Identify the Question Type

Grade 10 analytical geometry questions use coordinates on the Cartesian plane to calculate lengths, midpoints, and gradients. Every question begins with two or more given points — identify what information you have before choosing a formula.

Type 1: Distance between two points

Trigger words: "Calculate the length", "Find the distance", "How far apart"

Trigger structure: Two coordinate points are given. The question asks for the length of the segment between them.

Do not confuse with: Midpoint — the midpoint question asks for a coordinate, not a length.

Method (no numbers — just the steps)

Use the distance formula: d = √[(x₂ − x₁)² + (y₂ − y₁)²]
Substitute the coordinates carefully (mind the signs)
Simplify inside the root, then calculate
Leave in surd form unless told to round

Grade 10Paper 2Level 2 — Routine Procedures

2.1 A(1 ; 3) and B(4 ; 7) are two points. Calculate the length of AB.

Source: DBE Mathematics Paper 2, November 2021, Question 2.1

Common mistake

Squaring the differences first and then adding — correct. Adding first and then squaring — wrong.

See the progression — same type, increasing difficulty

Easy

Find the length of PQ if P(0 ; 0) and Q(3 ; 4).

Practice question — not sourced from a past paper.

Medium

Calculate the distance between M(−2 ; 3) and N(4 ; −1).

Practice question — not sourced from a past paper.

Hard

A(2 ; k) and B(6 ; 2) are such that AB = √32. Determine the value(s) of k.

Practice question — not sourced from a past paper.

Type 2: Midpoint of a line segment

Trigger words: "Find the midpoint", "M is the midpoint", "Determine the coordinates of the midpoint"

Trigger structure: Two endpoint coordinates are given. The question asks for the point halfway between them.

Do not confuse with: Distance — midpoint gives a coordinate pair (x ; y), not a length.

Method (no numbers — just the steps)

Use the midpoint formula: M = ((x₁ + x₂)/2 ; (y₁ + y₂)/2)
Add the x-coordinates and divide by 2 for the x-coordinate of M
Add the y-coordinates and divide by 2 for the y-coordinate of M

Grade 10Paper 2Level 1 — Knowledge

Find the midpoint of the segment joining A(2 ; 6) and B(8 ; 2).

Practice question — not sourced from a past paper.

See the progression — same type, increasing difficulty

Easy

M is the midpoint of P(0 ; 4) and Q(6 ; 0). Find the coordinates of M.

Practice question — not sourced from a past paper.

Medium

M(3 ; 1) is the midpoint of A(1 ; −3) and B. Find the coordinates of B.

Practice question — not sourced from a past paper.

Hard

A(p ; 2) and B(4 ; q) have midpoint M(1 ; −1). Determine the values of p and q.

Practice question — not sourced from a past paper.

Type 3: Gradient of a line

Trigger words: "Calculate the gradient", "Find m", "Are the lines parallel or perpendicular?"

Trigger structure: Two points or a line equation is given. The question asks for the slope or asks you to compare slopes.

Do not confuse with: The y-intercept — gradient is the m in y = mx + c, not the c.

Method (no numbers — just the steps)

Use m = (y₂ − y₁)/(x₂ − x₁)
For parallel lines: m₁ = m₂
For perpendicular lines: m₁ × m₂ = −1

Grade 10Paper 2Level 2 — Routine Procedures

A(−1 ; 4) and B(3 ; −2) are two points. Calculate the gradient of AB.

Practice question — not sourced from a past paper.

Common mistake

Inverting the formula — it is change in y divided by change in x, not the other way around.

See the progression — same type, increasing difficulty

Easy

Find the gradient of the line passing through (2 ; 1) and (5 ; 7).

Practice question — not sourced from a past paper.

Medium

Line AB has gradient 3. Line CD is perpendicular to AB. Write down the gradient of CD.

Practice question — not sourced from a past paper.

Hard

A(2 ; 3), B(5 ; k), and C(8 ; −1) are collinear. Determine the value of k.

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.