Geometry and Measurement — Free IB MYP4 Mathematics (Standard) Questions

1QuestionConstructing Triangles with Given ConditionsConcept Practice

4 marks~6 minCriterion A

A surveyor is planning a triangular plot of land. Two boundary lines meet at point

A

: one runs

5\ \text{cm}

(on the scale drawing) to point

B

, and the other runs at

60°

to point

C

, which is

7\ \text{cm}

from

A

.

a

Construct triangle

ABC

with

AB = 5\ \text{cm}

,

\angle BAC = 60°

, and

AC = 7\ \text{cm}

. Label all three vertices. [1]

b

Measure

BC

to the nearest millimetre and record your answer in centimetres. [1]

c

The surveyor states that the plot is valid only if all three boundary lengths are less than

6.5\ \text{cm}

. Advise the surveyor whether this plot meets the requirement, referring to each boundary length in your response. [2]

Solutions

2QuestionConstructing Perpendicular and Angle BisectorsConcept Practice

2 marks~3 minCriterion B

The diagrams below show a series of angle bisector constructions. The first diagram shows an angle of 60 degrees with one bisector dividing it into two 30-degree angles. The second diagram shows the same angle with two bisectors, dividing it into four 15-degree angles. The third diagram shows the angle with four bisectors, dividing it into eight 7.5-degree angles.

Outline the visual pattern in the number of bisectors as the construction continues. Predict how many bisectors the next diagram in the sequence would have.

Solutions

3QuestionMeasuring and Drawing BearingsConcept Practice

2 marks~3 minCriterion A

A coast guard vessel departs from port A. A north line is shown at A. Point B marks a lighthouse.

a

Determine the three-figure bearing of B from A. [1]

b

The coast guard must travel on a bearing of

045°

to reach a rescue zone C from A. Justify whether the lighthouse at B or the rescue zone at C is further clockwise from north when viewed from A. [1]

Solutions

4QuestionProperties 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

5QuestionTypes of Angles and Angle RelationshipsConcept 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 relationship between the number of sides and the sum of interior angles.

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

The architect needs a tile with a sum of interior angles of exactly

1080°

and is considering an octagonal tile.

a

Deduce the relationship between the number of sides

n

and the sum of interior angles. [1]

b

Justify whether an octagonal tile satisfies the architect's requirement. [1]

Solutions

6QuestionInterior and Exterior Angles of PolygonsConcept Practice

2 marks~3 minCriterion A

Architects designing a tiled floor use regular hexagonal tiles. For tiles to fit together without gaps, the interior angles at each vertex must sum to exactly

360°

.

Using the formula

S = (n-2) \times 180°

, where

n

is the number of sides, calculate the sum of the interior angles of a regular hexagon and hence find one interior angle. [1]

Justify whether a regular hexagon can tile a flat floor without gaps, using your calculated interior angle. [1]

Solutions

7QuestionUsing Congruent Triangles in ProofsConcept Practice

2 marks~3 minCriterion A

A structural engineer uses two triangular steel brackets to reinforce a bridge joint. The brackets are congruent, with

\triangle ABC \cong \triangle DEF

. In

\triangle ABC

, side

AB = 5

cm, side

BC = 7

cm, and

\angle ABC = 64°

.

a

Deduce the side in

\triangle DEF

that corresponds to side

AB

, and state its length. [1]

b

Justify whether the engineer can conclude that

\angle DEF = 64°

without taking any further measurements of

\triangle DEF

. [1]

Solutions

8QuestionAngles at the Centre and in the CircleConcept Practice

4 marks~6 minCriterion A

An architect is designing a circular skylight. Points

A

,

B

, and

C

lie on the circumference of a circle with centre

O

. The architect measures

\angle BAC = 34°

. The central angle

\angle BOC

is modelled by the expression

(2x + 10)°

.

a

Deduce the measure of

\angle BOC

. [1]

b

Find the value of

x

. [1]

c

The architect states: "Because the central angle exceeds $60°$ , arc $BC$ spans more than one-sixth of the full circle." Justify whether this statement is correct. [2]

Solutions

9QuestionUsing Compass and Ruler for Geometric ConstructionsAssessment Practice

2 marks~3 minCriterion C

A city planner must build a straight path through a rectangular park, connecting the midpoints of the two shorter sides so that the path is equidistant from both longer sides.

Figure 1 shows rectangle

ABCD

, where

AB

and

CD

are the shorter sides.

a

Construct the path using only a compass and ruler. Describe each step of your construction. [1]

b

The park is later found to be a trapezoid, not a rectangle. Advise the city planner whether this construction still meets the design requirement, justifying your answer with a geometric reason. [1]

Solutions

10QuestionConstructing Perpendicular and Angle BisectorsAssessment Practice

6 marks~9 minCriterion D

A self-driving boat navigates a narrow channel with an irregularly shaped island. Its sensors continuously identify the two nearest points on opposite sides of the channel — one on the island's perimeter, one on the channel wall — and the boat must remain equidistant from both to avoid collision. The channel is 50 m wide at its narrowest point.

a

Explain how perpendicular bisectors can model the boat's safe navigation path through the channel. [2]

b

Discuss two limitations of applying this perpendicular bisector model in a real-world maritime environment. [2]

c

The boat's sensors have a positional error of

\pm 1.5

m. Advise the boat's navigation engineers whether the perpendicular bisector path alone provides a sufficient safety margin at the 50 m narrowest point, and justify what additional measures, if any, are required for safe passage. [2]

Solutions

11QuestionCreating 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

12QuestionCreating and Interpreting Scale DiagramsAssessment Practice

6 marks~9 minCriterion B

A cartographer is creating maps of the same region at different scales. The table below shows the relationship between scale factor and the measured map distance between two landmarks.

Scale factor denominator	5 000	10 000	20 000	40 000
Map distance (cm)	20	10	5	2.5

The actual distance between the two landmarks remains constant throughout.

a

Analyse the relationship between the scale factor denominator and the map distance, identifying any pattern. [2]

b

Deduce the map distance between the landmarks on a map with scale factor 1:80 000, showing your reasoning clearly. [2]

c

A cartographer claims that a map distance of 8 cm is suitable for a printed map where the minimum legible distance between two labelled landmarks is 6 cm. Identify the scale factor that produces this 8 cm map distance and justify whether the cartographer's claim is valid. [2]

Solutions

13QuestionWord Problems Involving Bearings and MapsAssessment Practice

6 marks~9 minCriterion D

A pilot must fly from Airport A to Airport B, located 350 km away on a bearing of

055°

. A wind blows from

290°

at 50 km/h. The pilot maintains an airspeed of 280 km/h.

a

Construct a labelled vector diagram and calculate the required heading and ground speed so that the plane tracks directly to Airport B. [3]

b

Deduce the estimated flight time, in minutes, using your results from part (a). [1]

c

Advise the pilot on whether the single calculated heading of approximately

047°

and flight time of approximately 85 minutes can be relied upon for the flight plan, referring to at least two real-world factors and their effect on arrival time and position. [2]

Solutions

14QuestionTypes of Angles and Angle RelationshipsAssessment Practice

6 marks~9 minCriterion D

A ship navigates a narrow channel using triangulation. The captain measures the angle to landmark A as

35°

and the angle to landmark B as

72°

. The straight-line distance between A and B along the shore is

4.2 \text{ km}

.

a

Identify two sources of error in this triangulation method — one instrumental and one environmental — and explain how each may affect the accuracy of the ship's estimated position. [2]

b

Analyse how the geometric configuration of the triangle formed by the ship and the two landmarks affects the precision of the position fix, referring to the angles given. [2]

c

The captain considers switching to GPS, which has a typical positional accuracy of

\pm 5 \text{ m}

. Advise the captain on whether GPS should replace triangulation entirely for this channel crossing, justifying your recommendation with reference to both methods' limitations. [2]

Solutions

15QuestionPressure Force Area ApplicationsAssessment Practice

6 marks~9 minCriterion C

A hydraulic press transmits pressure from a small piston to a large piston. Pascal's principle states that pressure applied to an enclosed fluid is transmitted equally throughout:

P_2 = P_1

. The relationship between pressure, force, and area is

P = \dfrac{F}{A}

.

For three independent cases, the following data are given.

F_1

(N): 100, 250, 400

P_1

(Pa): 2000, 2500, 3200

F_2

(N): 5000, 15000, 28000

a

Calculate the area

A_1

of the small piston for each case. [2]

b

Deduce the area

A_2

of the large piston for each case, using Pascal's principle. [2]

c

A design engineer claims that doubling

F_1

while keeping

P_1

constant will always double

A_2

. Justify whether this claim is correct. [2]

Solutions

16QuestionMulti-Step Problems with Compound MeasuresAssessment Practice

5 marks~8 minCriterion A

Two cars complete a 500 km journey. Car A takes 5 hours and uses 40 litres of fuel. Car B takes 5.5 hours and uses 45 litres in total; however, Car B has a fuel leak, losing fuel at a constant rate of 2 litres per hour throughout the journey.

a

Calculate the fuel consumption rate of Car A in litres per 100 km. [1]

b

Deduce the volume of fuel Car B actually used for propulsion, excluding leaked fuel, and hence calculate Car B's fuel consumption rate in litres per 100 km. [3]

c

A car manufacturer claims that any vehicle achieving under 7.5 litres per 100 km is classified as fuel-efficient. Advise a cost-conscious buyer which car to purchase, justifying your recommendation using your results. [1]

Solutions

17QuestionMulti-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

18QuestionConverting Between Metric UnitsAssessment Practice

2 marks~3 minCriterion D

A municipal water authority bills customers in cubic metres (

\text{m}^3

). A community swimming pool holds exactly

23\,500

litres of water, and the billing rate is

\$2.40

per

\text{m}^3

. The authority's system truncates all volumes to the nearest whole

\text{m}^3

before calculating the charge.

Given that

1\,\text{m}^3 = 1\,000

litres:

a

Calculate the exact volume of the pool in cubic metres and the exact cost of filling it. [1]

b

Justify whether the authority should continue using truncation as its billing method, with reference to your calculated values. [1]

Solutions

19QuestionCriteria 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

20QuestionUsing Scale Factor in Similar FiguresAssessment Practice

4 marks~6 minCriterion D

A group of hikers plans a route using a topographic map with a scale of

1 : 50\,000

.

a

The distance between starting point

A

and campsite

B

measures

8.5

cm on the map. Calculate the actual distance in kilometres between

A

and

B

. [1]

b

The total elevation gain along the route from

A

to

B

is

400

m. Show that the average slope of the terrain, expressed as a percentage, is approximately

9.4\%

. [1]

c

A hiking guidebook classifies any route with an average slope above

10\%

as strenuous. Using your results from parts (a) and (b), advise the group of hikers whether the route from

A

to

B

should be classified as strenuous, and explain one real-world factor that could make the actual difficulty differ from this classification. [2]

Solutions

21QuestionUsing 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

22QuestionCyclic Quadrilaterals and ChordsAssessment Practice

6 marks~9 minCriterion B

An architect is designing a circular plaza. Four anchor points

A

,

B

,

C

,

D

lie on the circle, forming cyclic quadrilateral

ABCD

. A diagonal support beam runs along chord

AC

.

The following measurements are recorded:

AB

(cm): 5, 8, 7, 6, 5

BC

(cm): 5, 8, 7, 8, 12

\angle ADC

(degrees): 60, 120, 90, 60, 90

a

Deduce a general formula for the length of chord

AC

in terms of

AB

,

BC

, and

\angle ADC

. [3]

b

Calculate the length of chord

AC

for each set of measurements. [2]

c

The support beam

AC

can only be installed if its length is strictly less than

AB + BC

. Justify, for each case, whether the beam can be installed, and explain what geometric property guarantees this will always be the case. [1]

Solutions

23QuestionCyclic Quadrilaterals and ChordsAssessment Practice

4 marks~6 minCriterion D

A horticulturalist designs a circular greenhouse whose floor plan is modelled as a cyclic quadrilateral

ABCD

with side lengths

a

,

b

,

c

,

d

and semi-perimeter

s = \dfrac{a+b+c+d}{2}

. The area is estimated using Brahmagupta's formula:

\text{Area} = \sqrt{(s-a)(s-b)(s-c)(s-d)}

a

Explain why Brahmagupta's formula requires

ABCD

to be a cyclic quadrilateral, referring to a key angle property. [1]

b

Discuss one strength and one limitation of using this formula to estimate the usable growing area of a real greenhouse. [2]

c

The horticulturalist measures the constructed greenhouse and finds the actual dimensions differ from the design. Advise the horticulturalist whether Brahmagupta's formula remains a reliable basis for calculating growing area, and recommend one specific adjustment to the design process. [1]

Solutions

24QuestionAngles in the Same Segment and Alternate Segment TheoremAssessment Practice

4 marks~6 minCriterion C

A circular road junction is modelled by a circle with four points

A

,

B

,

C

, and

D

on its circumference, forming cyclic quadrilateral

ABCD

. A traffic engineer records the following angles:

\angle BAC = (3x + 10)^\circ, \quad \angle BDC = (5x - 30)^\circ, \quad \angle BCD = (4y)^\circ

a

Explain why

\angle BAC = \angle BDC

. [1]

b

Deduce two equations in

x

and

y

using the properties of cyclic quadrilaterals. [1]

c

Solve your equations to find

x

and

y

, showing all working. [1]

d

Calculate

\angle BAD

and justify whether the engineer's claim that the layout forms a valid cyclic quadrilateral is correct. [1]

Solutions

Geometry and Measurement — Free MYP4 Mathematics (Standard) Practice Questions