3.5 Perpendicular bisectors questions for IB Maths AI

47 questions on Perpendicular bisectors, written by Augment in the style of IB exam papers. Six of them are below in full, to answer on paper.

IB Mathematics: Applications and Interpretation
Part of 3 Geometry and trigonometry
SL and HL

Questions by paper and level

SL and HL questions are for students at either level. HL only questions cover what only HL students study.

Paper	SL and HL	HL only	All
Paper 1	21	1	22
Paper 2	23	0	23
Paper 3HL only	0	2	2
Every paper	44	3	47

Counted on 10 November 2026

Questions to try

Written by Augment in the style of IB exams, not past papers. Answer them on paper, then check each one against its markscheme with a free account.

Question 1 · AI-00109Paper 1 · 6 marks

A farming cooperative in the Rift Valley has two water wells at A $(1, 7)$ and B $(9, 3)$ , where distances are in kilometres. The cooperative wants to know which points are the same distance from both wells.

(a)
Find the coordinates of the midpoint of [AB].
2 marks
(b)
Find the gradient of [AB].
1 mark
(c)
Find the equation of the perpendicular bisector of [AB]. Give your answer in the form $y = m x + c$ .
2 marks
(d)
A straight road runs along the line $y = 7$ . Write down the coordinates of the point on the road that is equidistant from both wells.
1 mark

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Question 2 · AI-00223Paper 2 · 16 marks

The diagram shows the positions of three attractions in a theme park on a coordinate grid. The attractions are the Ark at A(0, 8), the Big Wheel at B(12, 12) and the Coaster at C(8, 0). The $y$ -axis points north and one unit represents 10 metres.

Coordinate grid from 0 to 14 on both axes, 1 unit = 10 m. Attraction A at (0, 8), B at (12, 12) and C at (8, 0).

(a)
Find the distance, in metres, between the Ark and the Big Wheel.
2 marks
(b)
Find the bearing of the Big Wheel from the Coaster.
3 marks
(c)
(i) Write down the coordinates of the midpoint of [AC].
(ii) Find the gradient of [AC].
(iii) Hence find the equation of the perpendicular bisector of [AC].
4 marks
(d)
The perpendicular bisector of [AB] has equation $y = - 3 x + 28$ .
A fountain F is to be built at the point that is equidistant from all three attractions. Find the coordinates of F.
2 marks
(e)
Find the distance, in metres, from the fountain to each attraction.
2 marks
(f)
Find the size of angle ABC.
3 marks

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Question 3 · AI-01800Paper 3 · HL only · 27 marks

The purpose of this question is to use Voronoi diagrams to help a city council decide where to build a new public library.

Mehmet is a planner for a city whose boundary is modelled by the rectangle $0 \leq x \leq 12$ , $0 \leq y \leq 10$ , where one unit represents 1 km. The city has five libraries, at $A (1, 3)$ , $B (7, 1)$ , $C (11, 5)$ , $D (7, 9)$ and $E (2, 9)$ . Residents are assumed to use the library nearest to their home.

The following diagram shows the libraries and part of the Voronoi diagram for the five libraries. The edge between the cells of $A$ and $B$ is missing.

Grid with 0 ≤ x ≤ 12, 0 ≤ y ≤ 10. Libraries A(1, 3), B(7, 1), C(11, 5), D(7, 9), E(2, 9). Voronoi edges drawn: (0, 6.25) to (4.5, 5.5); (4.5, 5.5) to (4.5, 10); (4.5, 5.5) to (5, 5); (5, 5) to (7, 5); (7, 5) to (12, 0); (7, 5) to (12, 10). The edge between A and B is missing.

(a)
Find the equation of the perpendicular bisector of $[A B]$ . Give your answer in the form $y = m x + c$ .
4 marks
(b)
Show that the point $(5, 5)$ is equidistant from $A$ , $B$ and $D$ .
3 marks
(c)
Hence find the coordinates of the end points of the missing edge.
2 marks
(d)
The council wants to build a new library at the point in the city that is furthest from its nearest existing library. A councillor claims that this point must be a vertex of the Voronoi diagram.
The vertices of the Voronoi diagram are $(4.5, 5.5)$ , $(5, 5)$ and $(7, 5)$ .
For each vertex, find the distance to its nearest library. Hence state which vertex the councillor would choose.
3 marks
(e)
By considering the corner $(12, 0)$ of the city, show that the councillor's claim is incorrect.
2 marks
(f)
Suggest one reason why the council might still prefer to build the new library at $(5, 5)$ .
1 mark
(g)
The council builds library $F$ at $(5, 5)$ .
(i) Write down the equation of the perpendicular bisector of $[F C]$ .
(ii) Find the equation of the perpendicular bisector of $[F B]$ .
3 marks
(h)
In the new Voronoi diagram, the cell of $F$ is a pentagon with vertices $(4, 2)$ , $(8, 4)$ , $(8, 6)$ , $(4.5, 7.75)$ and $G$ . The point $G$ is the intersection of the perpendicular bisectors of $[F A]$ and $[F E]$ , which have equations $y = - 2 x + 10$ and $3 x - 4 y = - 17.5$ .
Find the coordinates of $G$ .
2 marks
(i)
The area of a polygon with vertices $(x_{1}, y_{1}), (x_{2}, y_{2}), \dots, (x_{n}, y_{n})$ , listed in order around the polygon, is
$\frac{1}{2} ∣ (x_{1} y_{2} - x_{2} y_{1}) + (x_{2} y_{3} - x_{3} y_{2}) + \dots + (x_{n} y_{1} - x_{1} y_{n}) ∣ .$
Use this formula to find the area of the cell of $F$ .
4 marks
(j)
The population density of the city is approximately 1800 people per km².
Estimate the number of residents who will use library $F$ .
2 marks
(k)
State one reason why the actual number of people using library $F$ may be different from this estimate.
1 mark

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Question 4 · AI-00213Paper 1 · 7 marks

Two villages, Arroyo and Bellavista, are shown on a map with a coordinate grid where one unit represents one kilometre. Arroyo is at A(2, 3) and Bellavista is at B(10, 7).

The council wants to build a water tower that is the same distance from both villages.

(a)
Find the coordinates of the midpoint of [AB].
2 marks
(b)
Find the gradient of [AB].
2 marks
(c)
Find the equation of the perpendicular bisector of [AB]. Give your answer in the form $y = m x + c$ .
3 marks

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Question 5 · AI-00224Paper 2 · 16 marks

Two secondary schools, Palmer High at P(2, 9) and Quayside Academy at Q(12, 3), are shown on a map with a coordinate grid. One unit represents one kilometre.

(a)
Find the coordinates of the midpoint of [PQ].
2 marks
(b)
Find the equation of the perpendicular bisector of [PQ]. Give your answer in the form $a x + b y + d = 0$ , where $a, b, d \in Z$ .
4 marks
(c)
A straight road runs from the town hall at T(0, 1) along the line $y = 0.5 x + 1$ . A bus stop S is to be built on this road so that it is equidistant from the two schools.
Find the coordinates of S.
3 marks
(d)
Find the distance from S to Palmer High.
2 marks
(e)
Students go to the school nearer to their home. Determine which school a student living at H(10, 8) attends.
2 marks
(f)
A school bus travels from T to S along the road at an average speed of 36 km h⁻¹. Find the time taken, in minutes.
3 marks

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Question 6 · AI-01828Paper 3 · HL only · 26 marks

This question explores how logarithms and matrices can be used to locate an animal from the strength of a radio signal.

Batbayar is an ecologist studying snow leopards in the Altai mountains of Mongolia. One leopard wears a collar that sends out a radio signal. Three receiving towers, $T_{1}$ , $T_{2}$ and $T_{3}$ , are at the points $(0, 0)$ , $(10, 0)$ and $(4, 8)$ on a map, where units are kilometres.

The strength of the signal received by a tower, $P$ decibel-milliwatts (dBm), is modelled by

P = - 40 - 20 \log_{10} d,

where $d$ km is the distance from the collar to the tower. At one moment the three towers receive the following signal strengths.

Tower	$T_{1}$	$T_{2}$	$T_{3}$
$P$ (dBm)	−56.5	−54.0	−54.6

(a)
Show that $d = 10^{\frac{- 40 - P}{20}}$ , and hence find the distance from the collar to $T_{1}$ .
3 marks
(b)
Find the distance from the collar to $T_{2}$ and the distance from the collar to $T_{3}$ .
2 marks
(c)
Using the signals at $T_{1}$ and $T_{2}$ only, the collar lies on both of the circles
$x^{2} + y^{2} = d_{1}^{2} and (x - 10)^{2} + y^{2} = d_{2}^{2} .$
Use technology to find the two possible positions of the collar.
3 marks
(d)
Explain how the signal at $T_{3}$ shows which of these positions is correct.
1 mark
(e)
To avoid solving non-linear equations, Batbayar uses the third circle, $(x - 4)^{2} + (y - 8)^{2} = d_{3}^{2}$ , as well.
By subtracting equations, show that
$20 x = 100 + d_{1}^{2} - d_{2}^{2} and 8 x + 16 y = 80 + d_{1}^{2} - d_{3}^{2} .$
2 marks
(f)
The equations in part (e) can be written as $M (\begin{array}{c} x \\ y \end{array}) = b$ .
Write down the matrix $M$ and find the vector $b$ .
2 marks
(g)
Use $M^{- 1}$ to find the position of the collar.
3 marks
(h)
Show that $\det M \neq 0$ . Explain why this method would fail if the three towers were in a straight line.
2 marks
(i)
Batbayar notices that if $d_{1} = d_{2}$ , the first equation in part (e) becomes $x = 5$ .
State the geometrical meaning of the line $x = 5$ in relation to the towers $T_{1}$ and $T_{2}$ , and explain why the collar must lie on it in this case.
2 marks
(j)
Each tower measures signal strength to within 0.5 dBm of the true value.
Find the largest percentage error in the distance from the collar to $T_{2}$ that this could cause.
3 marks
(k)
A GPS reading later shows that the leopard was at $(6, 3)$ at that moment.
Find the distance between this position and the position found in part (g), giving your answer in metres.
2 marks
(l)
Suggest one reason why the model for the signal strength may be less reliable in the mountains.
1 mark

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