Geometry and Measurement — Free IB MYP5 Mathematics (Standard) Questions

1QuestionConstructing Perpendicular and Angle BisectorsConcept Practice

1 mark~2 minCriterion A

Classify each statement about the angle bisector in the diagram as True or False. Write your answers in the table below.

Angle: ∠AOB = 80°, with ray OC bisecting it, so that ∠AOC = ∠COB = 40°.

Statement Table:

Statement	∠AOC = ∠COB	∠AOB = 2 × ∠AOC	∠AOB = ∠AOC	∠COB = 40°
Classification	True or False	True or False	True or False	True or False

Solutions

2QuestionProperties of Regular and Irregular PolygonsConcept Practice

2 marks~3 minCriterion C

A landscape architect is designing a pavilion with a regular hexagonal floor plan. Each interior angle of the floor must be equal for the tiling to fit correctly.

a

Calculate the sum of the interior angles of a hexagon. [1]

b

The architect claims each interior angle measures 130°. Justify whether this claim is correct. [1]

Solutions

3QuestionInterior and Exterior Angles of PolygonsConcept Practice

2 marks~3 minCriterion A

A landscape architect is designing a public courtyard floor using a single type of regular polygon as the tile shape.

a

Explain why a regular hexagon can tile a floor without gaps or overlaps, referring to its interior angle. [1]

b

Discuss one limitation of restricting the tile design to a single type of regular polygon. [1]

Solutions

4QuestionUsing Scale Factor in Similar FiguresConcept Practice

2 marks~3 minCriterion A

An architect creates a scale model of a triangular roof truss. The model triangle

\triangle ABC

has side lengths

AB = 3\ \text{cm}

,

BC = 4\ \text{cm}

, and

AC = 5\ \text{cm}

. The actual truss is represented by the similar triangle

\triangle DEF

, with corresponding side lengths

DE = 60\ \text{cm}

,

EF = 80\ \text{cm}

, and

DF = 100\ \text{cm}

.

Deduce the scale factor

k

from

\triangle ABC

to

\triangle DEF

, and interpret what this value means for the architect's model. [2]

Solutions

5QuestionAngles in the Same Segment and Alternate Segment TheoremConcept Practice

2 marks~3 minCriterion A

A satellite dish engineer checks that a circular signal reflector is correctly aligned. The circle has centre

O

. Points

A

,

B

, and

C

lie on the circumference. The engineer measures

\angle ABC = 32°

.

a

Calculate

\angle AOC

. [1]

b

The alignment specification requires the central angle

\angle AOC

to be greater than

60°

. Justify whether the reflector meets this specification, using your result from part (a). [1]

Solutions

6QuestionConstructing Perpendicular and Angle BisectorsAssessment Practice

3 marks~5 minCriterion B

Investigate whether the angle bisectors of the interior angles of a regular polygon always intersect at a single point. Consider a regular pentagon (5 sides), a regular hexagon (6 sides), and a regular heptagon (7 sides). For each polygon, draw the angle bisectors of all interior angles.

a

For the regular pentagon, do the angle bisectors all meet at one point? If yes, describe that point's location relative to the polygon.

b

Based on the results for the pentagon, hexagon, and heptagon, state a general rule about the angle bisectors of any regular polygon.

c

Test your rule by predicting whether the angle bisectors of a regular octagon (8 sides) will all intersect at a single point. Explain your reasoning.

Solutions

7QuestionConstructing Perpendicular and Angle BisectorsAssessment Practice

4 marks~6 minCriterion C

Consider a line segment

AB

drawn on a grid.

Explain, step-by-step, how to construct the perpendicular bisector of line segment

AB

using a compass and straightedge. Your explanation must include:

a

A sketch of the construction, clearly labeling points

A

,

B

,

C

,

D

, and

M

, where

C

and

D

are the intersection points of the arcs, and

M

is the midpoint of

AB

.

b

A detailed written description of each step in the construction, including the compass settings used in each step.

c

A justification for why the line segment

CD

is the perpendicular bisector of

AB

, referring to properties of congruent triangles.

Solutions

8QuestionConstructing Perpendicular and Angle BisectorsAssessment Practice

6 marks~9 minCriterion D

Landscape designer Zara is planning a garden path that must be equidistant from two fixed points,

A

and

B

, marked on a scale drawing. The line segment

AB

is drawn below.

a

Construct the perpendicular bisector of

AB

using a compass and straightedge only. All construction arcs must be visible. [2]

b

Label a point

P

on the perpendicular bisector. Measure and record the distances

PA

and

PB

in centimetres. [1]

c

Calculate the percentage difference between

PA

and

PB

using:

\text{Percentage Difference} = \frac{|PA - PB|}{\dfrac{PA + PB}{2}} \times 100

[1]

d

Zara's design specification states the path must be within 5\% of equidistant from

A

and

B

to be approved. Advise Zara whether her construction meets the specification, and explain what any deviation from 0\% reveals about the accuracy of the compass-and-straightedge method. [2]

Solutions

9QuestionCreating and Interpreting Scale DiagramsAssessment Practice

6 marks~9 minCriterion D

A nautical chart has a scale of 1:50 000. A narrow channel measures 2 cm wide on the chart, representing an actual width of 1 000 m. The chart has been exposed to moisture and has uniformly shrunk by 2%.

a

Calculate the new width of the channel on the shrunken chart. [2]

b

Deduce the new scale of the shrunken chart, expressing your answer in the form

1:n

, and calculate the percentage error in the scale compared to the original. [2]

c

The ship is 900 m wide. Advise the captain whether the shrunken chart can be used safely to navigate the channel, justifying your advice with calculations of the actual clearance on each side of the ship. [2]

Solutions

10QuestionCreating and Interpreting Scale DiagramsAssessment Practice

4 marks~6 minCriterion B

A cartographer is creating maps of the same region at three different scales. The base map is a square with side length 1 unit, representing a real-world area of 1 km

^2

.

a

Construct a table showing the Map Area (units

^2

) and Real-World Area (km

^2

) for scale factors

k = 1, 2, 3, 4, 5

. Show the calculation for each Map Area. [2]

b

Deduce a general formula for the Real-World Area in terms of scale factor

k

. [1]

c

A regional planning authority requires a map that represents at least 20 km

^2

. Advise the cartographer whether a scale factor of 4 is sufficient to meet this requirement, and state the minimum integer scale factor that does. [1]

Solutions

11QuestionWord Problems Involving Bearings and MapsAssessment Practice

4 marks~6 minCriterion A

A ship departs from port A on a bearing of

075°

and travels 80 km to port B. It then sails from port B on a bearing of

160°

and travels 60 km to port C.

a

Construct a clearly labelled diagram of the ship's journey, showing ports A, B, and C, all bearings, distances, and north directions. [1]

b

Deduce the distance AC, giving your answer to three significant figures. [2]

c

A coastguard at port A considers any vessel within 115 km and on a bearing between

100°

and

120°

to be within its patrol zone. Advise the coastguard whether port C requires monitoring, justifying your answer with both conditions. [1]

Solutions

12QuestionUnderstanding Bearings in NavigationAssessment Practice

6 marks~9 minCriterion D

A search and rescue team uses bearings to locate a lost hiker. Station A and Station B are 8 km apart, with Station A directly west of Station B. Station A detects the hiker on a bearing of

030°

and Station B detects the hiker on a bearing of

330°

.

a

Deduce the size of each interior angle in triangle

ABH

, where

H

is the hiker's estimated location. [2]

b

Calculate the distance from Station A to the hiker's estimated location. [2]

c

The rescue team deploys a helicopter when the hiker is more than 3.5 km from Station A, and a ground team otherwise. Advise the rescue coordinator which option to deploy, and explain one reason why the mathematical model may make this advice unreliable. [2]

Solutions

13QuestionProperties of Regular and Irregular PolygonsAssessment Practice

2 marks~3 minCriterion B

A landscape architect is designing decorative paving tiles, each shaped as a regular polygon. The table below shows the sum of interior angles for polygons with 3 to 8 sides.

Number of sides	3	4	5	6	7	8
Sum of interior angles (°)	180	360	540	720	900	1080

The architect needs a tile shaped as a regular decagon (10 sides).

a

Deduce the general rule for the sum of interior angles of a polygon with

n

sides. [1]

b

The architect states: "A regular decagon tile will have interior angles greater than 135°, making it unsuitable for corner paving." Justify whether this statement is correct. [1]

Solutions

14QuestionParallel Lines and TransversalsAssessment Practice

4 marks~6 minCriterion D

A road engineer is designing two parallel lanes of a highway intersected by a slip road acting as a transversal. The angle formed between the slip road and the near lane is

(2x + 10)°

. The corresponding angle at the far lane is

(3x - 5)°

.

a

Deduce an equation relating the two angles and solve for

x

. [2]

b

Justify whether the slip road crosses both lanes at an acute or obtuse angle, using the calculated angle measure. [1]

c

The highway design requires that no approach angle exceeds

50°

for driver safety. Advise the engineer whether this design should be approved. [1]

Solutions

15QuestionMulti-Step Problems with Compound MeasuresAssessment Practice

6 marks~9 minCriterion B

A manufacturer produces solid metal cubes of increasing side lengths, all from the same material. The side lengths and measured masses are recorded below.

Side length

s

(cm): 1, 2, 3, 4, 5

Mass

M

(g): 2, 8, 18, 32, 50

a

Analyse the mass values to identify the relationship between

M

and

s

. Show the first and second differences between consecutive mass values and justify why the relationship is quadratic. [2]

b

Deduce a formula for the density

D

(g/cm³) of a cube in terms of its side length

s

, using

D = \dfrac{M}{V}

. Show all working. [2]

c

The manufacturer requires a cube of side length 6 cm to have a density no greater than 0.30 g/cm³ to meet a safety specification. Using your formula from part (b), advise the manufacturer whether this cube meets the specification and recommend one practical course of action if it does not. [2]

Solutions

16QuestionConverting Between Metric UnitsAssessment Practice

3 marks~5 minCriterion A

A rectangular garden measures 550 cm by 320 cm. A landscaper plans to cover the entire surface with topsoil costing 5 dollars per

\text{m}^2

and fence the perimeter at 15 dollars per metre.

a

Calculate the area of the garden in

\text{m}^2

. [1]

b

Calculate the total cost of the topsoil and the total cost of the fencing, showing all unit conversions. [1]

c

The landscaper has a budget of 350 dollars. Justify whether this budget is sufficient to complete both the topsoil and fencing. [1]

Solutions

17QuestionConverting Between Metric UnitsAssessment Practice

6 marks~9 minCriterion C

A satellite image analyst measures features at three scales. Each measurement is the side length of a square region.

Square A: side

= 1 \ \text{mm}

Square B: side

= 1 \ \text{cm}

Square C: side

= 1 \ \text{dm}

a

Calculate the area of each square in

\text{mm}^2

,

\text{cm}^2

, and

\text{m}^2

. [3]

b

A colleague claims: "To convert an area measurement, you use the same conversion factor as for length." Using your results from part (a), deduce the correct general rule for converting between area units, and identify the error in the colleague's claim. [2]

c

Justify why area unit conversions require the linear conversion factor to be squared, relating your reasoning to the definition of area as a two-dimensional measure. [1]

Solutions

18QuestionConverting Between Metric UnitsAssessment Practice

6 marks~9 minCriterion D

A land surveyor uses a high-precision laser distance meter to measure a rectangular plot intended for a community garden.

Length:

54\,500 \text{ cm}

Width:

38\,200 \text{ cm}

The manufacturer specifies an accuracy of

\pm 0.05\%

for each measurement. Recall that

1 \text{ ha} = 10\,000 \text{ m}^2

.

a

Show that the area of the plot is approximately

20.819 \text{ ha}

. Show all working, including unit conversions. [2]

b

Deduce the possible range of the area, in hectares, accounting for the

\pm 0.05\%

measurement error. [2]

c

Advise the surveyor whether hectares, square metres, or square centimetres is the most appropriate unit for reporting the area of this community garden, justifying your recommendation with reference to both the scale of the area and the precision of the measurements. [2]

Solutions

19QuestionUsing Scale Factor in Similar FiguresAssessment Practice

6 marks~9 minCriterion B

A designer produces a series of similar pentagonal prisms for architectural models. The table below summarises measurements for the first three prisms in the series.

Side length (cm)	2	4	6
Perimeter of base (cm)	10	20	30
Area of base (cm²)	6.88	27.52	61.92
Volume (cm³)	34.40	275.20	928.80

In each prism, the height equals the side length.

a

Deduce a formula for the perimeter

P

of the pentagonal base in terms of its side length

s

. [1]

b

Deduce a formula for the area

A

of the pentagonal base in terms of

s

, showing that the ratio

\dfrac{A}{s^2}

is constant. [2]

c

Using your results from (a) and (b), deduce a formula for the volume

V

of the prism in terms of

s

. [1]

d

A client requests a prism with side length 8 cm, but the display cabinet has a maximum volume of 850 cm³. Calculate the volume of this prism and advise the client whether the prism can be displayed in the cabinet, justifying your answer. [2]

Solutions

20QuestionCriteria for Triangle Congruence SSS SAS ASA RHSAssessment Practice

6 marks~9 minCriterion C

A structural engineer is checking whether two triangular roof trusses,

\triangle ABC

and

\triangle DEF

, are identical in shape and size before installation. She records measurements progressively.

a

She finds

AB = DE

and

BC = EF

. Explain why this information alone is insufficient to confirm the trusses are congruent. [2]

b

She then confirms

\angle ABC = \angle DEF

. Justify why

\triangle ABC \cong \triangle DEF

can now be concluded, naming the congruence criterion used. [2]

c

A colleague suggests that knowing only

\angle ABC = \angle DEF

and

BC = EF

(without

AB = DE

) would be equally sufficient. Advise the engineer whether this measurement combination is reliable for confirming congruence, and state what additional measurement she would need. [2]

Solutions

21QuestionCriteria for Triangle Congruence SSS SAS ASA RHSAssessment Practice

2 marks~3 minCriterion D

Prefabricated roof trusses are manufactured to precise dimensions so that the SSS congruence criterion can confirm all trusses are identical, ensuring uniform load distribution across a structure.

Discuss one limitation of relying on the SSS congruence criterion to verify that manufactured roof trusses are congruent, and justify whether this limitation poses a genuine risk to structural safety after installation. [2]

Solutions

22QuestionAngles at the Centre and in the CircleAssessment Practice

5 marks~8 minCriterion C

A clock face has centre

O

and radius

r

. At 2:00 PM the tip of the hour hand is at point

A

; at 5:00 PM it has moved to point

B

, both points lying on the circumference. Point

C

also lies on the circumference of the clock face.

a

Calculate the angle

\theta = \angle AOB

, in degrees, swept by the hour hand between 2:00 PM and 5:00 PM. [1]

b

Given that

C

lies on the minor arc

AB

, deduce the angle

\angle ACB

. [2]

c

A clock designer claims that any position of

C

on the major arc

AB

gives an obtuse

\angle ACB

. Justify whether this claim is correct. [2]

Solutions

23QuestionAngles at the Centre and in the CircleAssessment Practice

4 marks~6 minCriterion B

A circular window in an architect's design is divided into four sections by a cyclic quadrilateral

ABCD

, where each vertex lies on the circle. The architect claims the design has structural symmetry only if opposite angles are supplementary.

Given that

\angle DAB = 73°

and

\angle BCD = x°

, study the diagram.

a

State the theorem that relates opposite angles in a cyclic quadrilateral. [1]

b

Calculate the value of

x

. [1]

c

Justify, using the relationship between inscribed angles and their subtended arcs, why opposite angles in any cyclic quadrilateral must always be supplementary, and hence assess whether the architect's symmetry condition is always satisfied. [2]

Solutions

24QuestionAngles in the Same Segment and Alternate Segment TheoremAssessment Practice

2 marks~3 minCriterion D

A surveyor places two observation points, A and B, on what is assumed to be a circular arc around the base of a distant communication tower. Using a theodolite, the surveyor measures the angle to the tower from each point, applying the angles in the same segment theorem to determine the tower's position.

Justify whether uneven terrain at the survey site undermines the reliability of the tower's calculated position. [2]

Solutions

Geometry and Measurement — Free MYP5 Mathematics (Standard) Practice Questions