Three towns, $A$ , $B$ and $C$ are represented as coordinates on a map. Town A is located at $(-6, -1)$ and town B is located at $(8, 6)$ . A straight highway runs exactly along the perpendicular bisector of the line segment [AB]. Find the coordinates of the midpoint of [AB]. Find the equation of the line that the highway follows. Town C is located on the highway and has an $x$ -coordinate of $-6$ . Find the $y$ -coordinate of Town C.

Midpoint $\left( \frac{-6+8}{2}, \frac{-1+6}{2} \right) = \left( 1, 2.5 \right)$ . (M1)A1 Gradient AB $= \frac{6 - (-1)}{8 - (-6)} = \frac{7}{14} = 0.5$ . (A1) Perpendicular gradient $= -2$ . $y - 2.5 = -2(x - 1) \implies y = -2x + 2 + 2.5 \implies y = -2x + 4.5$ . (M1)A1 Substitute $x = -6$ : $y = -2(-6) + 4.5 = 12 + 4.5 = 16.5$ . A1

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IB Maths AI SL · Unit 3: Geometry and Trigonometry

IB Maths AI SL Coordinate Geometry Questions

Updated October 2026

Exam-style IB Maths AI SL coordinate geometry questions with worked solutions. Start with the 5 fully worked examples below — each is solved step by step, showing the working that earns the marks — then try the practice questions and check your working against the mark scheme.

31 questions
Paper 1: 18
Paper 2: 13
7 easy
9 medium
8 hard
7 very hard
5 worked examples

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What you need to know

SL AI's geometry is applied: which cereal box has the lowest surface-area-to-volume ratio? Master the volume + surface-area formulas for cylinders, cones, and truncated shapes. 3D geometry — volume, surface area, and packaging overview →

What's examined in AI SL coordinate geometry

The question bank covers these coordinate geometry question types (number of questions in brackets):

Voronoi Diagrams (13)
Geometric Calculations (11)
Line Properties & Equations (7)

Coordinate Geometry worked examples

Worked example 1: Gradient between two points · easy

Find the gradient of the line through $A (- 3, 7)$ and $B (5, - 9)$ .

Solution

3 steps · 1 to fill in

Examiner tip: Keep the ORDER consistent between numerator and denominator — either point can be

(x_{1}, y_{1})

Worked example 2: Equation of a line from point and gradient · medium

A line has gradient $3$ and passes through $(2, 5)$ . Find its equation in $y = m x + c$ form.

Solution

3 steps

Examiner tip: Read the requested form carefully — sometimes exams want

a x + b y + d = 0

Worked example 3: x-intercept of a parallel line · hard

$L_{1}$ : $4 x + 2 y = 10$ . $L_{2}$ is parallel to $L_{1}$ and passes through $(1, 6)$ . Find the x-intercept of $L_{2}$ .

Solution

4 steps · 1 to fill in

Examiner tip: Always rearrange to

y = m x + c

before reading the gradient.

Worked example 4: Perpendicular gradient · easy

Line $L$ : $3 x - 5 y = 15$ . Find the perpendicular gradient.

Solution

3 steps · 1 to fill in

Examiner tip: Flip the fraction and change the sign.

Worked example 5: Perpendicular bisector as a boundary · medium

Towns at $A (2, 4)$ and $B (8, 16)$ . Find the equidistant boundary line in $y = m x + c$ form.

Solution

5 steps · 1 to fill in

Examiner tip: Use the MIDPOINT for a perpendicular bisector, not

A

B

Try these IB Maths AI SL coordinate geometry questions

Three questions from the bank, easiest first. Mark schemes and AI marking of your written working are in the practice area. One of them has its full worked solution here, open to everyone.

Question 1 · easy · 4 marks · Paper 1

The distance between two points with coordinates $(x_{1}, y_{1})$ and $(x_{2}, y_{2})$ is equal to $\sqrt{(x_{2} - x_{1})^{2} + (y_{2} - y_{1})^{2}}$ . Consider the points $A (1, - 2)$ and $B (40, - 100)$ .

Calculate the exact distance between points A and B.
Find the exact coordinates of the midpoint of the line segment [AB].

Attempt it and see the mark scheme →

Question 2 · medium · 6 marks · Paper 2

Three towns, $A$ , $B$ and $C$ are represented as coordinates on a map. Town A is located at $(- 6, - 1)$ and town B is located at $(8, 6)$ . A straight highway runs exactly along the perpendicular bisector of the line segment [AB].

Find the coordinates of the midpoint of [AB].
Find the equation of the line that the highway follows.
Town C is located on the highway and has an $x$ -coordinate of $- 6$ . Find the $y$ -coordinate of Town C.

Full worked solution Free, no sign-in

Midpoint $(\frac{- 6 + 8}{2}, \frac{- 1 + 6}{2}) = (1, 2.5)$ . (M1)A1
Gradient AB $= \frac{6 - (- 1)}{8 - (- 6)} = \frac{7}{14} = 0.5$ . (A1)
Perpendicular gradient $= - 2$ .
$y - 2.5 = - 2 (x - 1) ⟹ y = - 2 x + 2 + 2.5 ⟹ y = - 2 x + 4.5$ . (M1)A1
Substitute $x = - 6$ : $y = - 2 (- 6) + 4.5 = 12 + 4.5 = 16.5$ . A1

Every question here has a full worked solution. See them all →

Question 3 · hard · 7 marks · Paper 2

The equation of a straight coastline is modelled by the line $L_{1} : 2 y - x - 10 = 0$ . A boat is anchored at point $M (8, 18)$ . The coastguard needs to find the shortest distance from the boat to the coastline.

Find the gradient of the coastline $L_{1}$ .
Find the equation of the line $L_{2}$ , which passes through $M$ and is perpendicular to $L_{1}$ .
Find the coordinates of point $D$ , the intersection of $L_{1}$ and $L_{2}$ .
Calculate the shortest distance from the boat to the coastline.

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FAQ

How many IB Maths AI SL coordinate geometry questions are there?

There are 31 exam-style coordinate geometry questions in the AI SL question bank (Paper 1: 18 · Paper 2: 13), graded 7 easy, 9 medium, 8 hard, 7 very hard. Every question has a full IB-style mark scheme (M, A and R marks).

Is coordinate geometry on Paper 1 or Paper 2?

Both. In the bank, Paper 1: 18 · Paper 2: 13. Practise with your GDC — AI papers expect calculator methods throughout.

Where can I get the mark schemes?

Open the AI SL Unit 3 practice page: every question has a step-by-step IB-style mark scheme, and you can photograph your working for instant AI marking. The worked examples on this page are free.

What else is free here

Save your answers and track your progressA free account remembers the questions you have done on every topic and brings you back to this page.

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Teaching this topic?

Free ready-to-teach lesson (slides and teacher notes): Parallel & perpendicular lines.

IB Maths AI SL Coordinate Geometry Questions

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What you need to know

What's examined in AI SL coordinate geometry

Coordinate Geometry worked examples

Worked example 1: Gradient between two points · easy

Solution

Worked example 2: Equation of a line from point and gradient · medium

Solution

Worked example 3: x-intercept of a parallel line · hard

Solution

Worked example 4: Perpendicular gradient · easy

Solution

Worked example 5: Perpendicular bisector as a boundary · medium

Solution

Try these IB Maths AI SL coordinate geometry questions

Question 1 · easy · 4 marks · Paper 1

Question 2 · medium · 6 marks · Paper 2

Full worked solution Free, no sign-in

Question 3 · hard · 7 marks · Paper 2

Revise coordinate geometry for the exam

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