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

1QuestionInterior 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

2QuestionDensity Mass and VolumeConcept Practice

2 marks~3 minCriterion A

A furniture maker tests a rectangular block of oak before use in a project. The block has the following dimensions:

Length: 20 cm
Width: 10 cm
Height: 3 cm

The mass of the block is 480 g. Oak suitable for the project must have a density strictly greater than 0.75 g/cm³.

\text{Density} = \frac{\text{Mass}}{\text{Volume}}

Calculate the density of the block, then advise the furniture maker whether this block of oak should be used in the project. Justify your answer. [2]

Solutions

3QuestionAngles 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

4QuestionConstructing Perpendicular and Angle BisectorsAssessment Practice

6 marks~9 minCriterion B

Points

A(x_1, y_1)

and

B(x_2, y_2)

lie in the Cartesian plane. Their midpoint is

M

and the perpendicular bisector of

AB

passes through

M

.

Four cases are given.

A(0,0),\ B(2,2)

:

M=(1,1)

, perpendicular bisector

y = -x+2

A(1,2),\ B(3,4)

:

M=(2,3)

, perpendicular bisector

y = -x+5

A(-1,1),\ B(1,5)

:

M=(0,3)

, perpendicular bisector

y = -0.5x+3

A(2,-1),\ B(4,3)

:

M=(3,1)

, perpendicular bisector

y = -0.5x+2.5

a

Analyse the relationship between the slope of

AB

, the slope of its perpendicular bisector, and the coordinates of

M

in each case. [2]

b

Deduce a general equation for the perpendicular bisector of

AB

in the form

y = mx + c

, where

m

and

c

are expressed in terms of

x_1,\ y_1,\ x_2,\ y_2

. Show your algebraic reasoning clearly. [3]

c

A structural engineer checks that a support cable is equidistant from two anchor points

A(3,-2)

and

B(7,4)

. The cable must lie along the perpendicular bisector of

AB

. Apply your formula from part (b) to find the equation of this line, then justify whether the point

(5,1)

lies on the cable's path. [1]

Solutions

5QuestionConstructing Triangles with Given ConditionsAssessment Practice

4 marks~6 minCriterion C

A city planner is repositioning a triangular green space. On a coordinate grid, triangle

ABC

has vertices

A(1, 2)

,

B(4, 2)

, and

C(3, 5)

. After repositioning, triangle

A'B'C'

has vertices

A'(4, 6)

,

B'(7, 6)

, and

C'(6, 9)

.

a

Calculate the translation vector

\mathbf{v} = \begin{pmatrix} \Delta x \\ \Delta y \end{pmatrix}

that maps

A

onto

A'

. [1]

b

Verify that

\mathbf{v}

also maps

B

onto

B'

and

C

onto

C'

. [1]

c

The planner states: "The repositioned green space covers a completely different area of the city." Justify whether this statement is mathematically supported. [2]

Solutions

6QuestionConstructing Triangles with Given ConditionsAssessment Practice

6 marks~9 minCriterion D

Surveyors use triangulation to map a triangular field. Two sides measure

150 \text{ m}

and

200 \text{ m}

, with an included angle of

60°

. The surveying instrument has a measurement uncertainty of

\pm 0.5°

.

a

Calculate the area of the triangular field using the measured values. [2]

b

Determine the maximum and minimum possible areas of the field, given the uncertainty in the angle measurement. Show all working. [2]

c

The surveying company must report the field area to within

\pm 100 \text{ m}^2

of the true value to satisfy a legal land registration requirement. Advise the surveying company whether the instrument is sufficiently precise for this purpose, justifying your answer with reference to your results from parts (a) and (b). [2]

Solutions

7QuestionShading Regions Using Multiple LociAssessment Practice

6 marks~9 minCriterion A

A rectangular field is used for a treasure hunt. The treasure's location is determined by distance constraints from three landmarks: a tree at point

A

, a rock at point

B

located

8

m from

A

, and a straight fence running along one edge of the field.

a

Construct the locus of points exactly

5

m from

A

, and shade the region within

5

m of

A

. [1]

b

The treasure is also within

4

m of point

B

. Construct both circular loci on the same diagram and shade the region satisfying both constraints simultaneously. [2]

c

The treasure must additionally be within

3

m of the fence. Add this locus to your diagram, shade the region satisfying all three constraints, and justify whether the three constraints are sufficient to uniquely locate the treasure. [3]

Solutions

8QuestionCreating 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

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

10QuestionUnderstanding 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

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

12QuestionParallel Lines and TransversalsAssessment Practice

2 marks~3 minCriterion C

Elm Street and Oak Street are parallel. Maple Avenue crosses both streets as a transversal, forming eight angles at the two intersections. At the intersection with Elm Street, the angle on the upper-right measures

65°

.

a

State the measure of the corresponding angle at the intersection with Oak Street. [1]

b

A city planner claims that Maple Avenue crosses both streets "at the same angle," using this as evidence that the streets are parallel. Justify whether this claim constitutes valid geometric evidence of parallelism. [1]

Solutions

13QuestionInterior and Exterior Angles of PolygonsAssessment Practice

12 marks~18 minCriterion B

A designer is tiling a floor using identical regular polygon tiles that fit together edge-to-edge with no gaps. For tiles to tessellate, the interior angle must divide exactly into 360°.

The interior angle of a regular

n

-sided polygon is given by

A = \frac{180(n-2)}{n} \text{ degrees}

and the sum of interior angles by

S = 180(n-2)

degrees.

a

Calculate the sum of interior angles of a regular 20-sided polygon. [2]

b

Deduce a formula for the exterior angle

E

of a regular

n

-sided polygon in terms of

n

, showing your reasoning clearly. [3]

c

A tile has an interior angle of 150°. Calculate the number of sides of this tile. [3]

d

Justify whether the 150° tile can tessellate a floor on its own, using your results from parts (b) and (c). [4]

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

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

17QuestionConverting 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

18QuestionCriteria 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

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

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

21QuestionReal-Life Applications of Similarity and CongruenceAssessment Practice

4 marks~6 minCriterion A

A surveyor needs to find the height of a tall building. She stands a pole of height 3 m vertically near the building. The pole casts a shadow of 2 m, and the building casts a shadow of 30 m. The sun's rays strike both objects at the same angle, forming two similar triangles.

a

Construct a clearly labelled diagram showing the two similar triangles formed by the pole, the building, and their shadows. Label all known lengths and indicate the equal angles. [1]

b

Deduce a proportion relating the height of the pole, the length of the pole's shadow, the height of the building

h

, and the length of the building's shadow. [1]

c

Solve your proportion from part (b) to find

h

. Show all working. [1]

d

The building's shadow measurement carries an uncertainty of

\pm 0.5

m. Advise the surveyor whether this shadow measurement is precise enough to report the building's height reliably for a construction project requiring accuracy to the nearest metre. [1]

Solutions

22QuestionLength of arc and chord, perimeter and area of sector and segmentAssessment Practice

8 marks~12 minCriterion B

The table shows arc length and sector area values for four sectors, each from a circle of radius 10 cm.

Central angle (degrees)	30	60	90	120
Arc length (cm)	5.24	10.47	15.71	20.94
Sector area (cm²)	26.18	52.36	78.54	104.72

a

Describe the relationship between the central angle

\theta

and the arc length

L

, referring to at least two values in the table. [2]

b

Using the pattern in the table, write down a formula for the arc length

L

of a sector with central angle

\theta

degrees and radius

r

cm. Hence find the arc length of a sector with central angle

150^\circ

in the same circle (radius 10 cm). Give your answer in exact form and correct to 1 decimal place. [3]

c

A different circle has radius

r

cm. Two sectors of this circle have central angles

\alpha

degrees and

\beta

degrees respectively, where

\alpha + \beta = 180

. Show that the sum of their arc lengths equals

\pi r

, and justify why this result holds regardless of the individual values of

\alpha

and

\beta

. [3]

Solutions

23QuestionLength of arc and chord, perimeter and area of sector and segmentAssessment Practice

10 marks~15 minCriterion D

A city planner is designing a grass-covered traffic roundabout. Each roundabout is modelled as a sector of a circle with central angle

60°

. The diagram shows a sector with two radii of length

r

metres and the arc.

Sod (grass turf) costs 4.50 USD per square metre. The sector area formula is

A = \frac{\pi r^2}{6}.

The table below shows the installed radius and actual measured area at three sites.

Installed radius (m)	11.5	12.0	12.5
Actual area (m²)	68.00	75.00	82.50

a

Calculate the predicted area

A

for each installed radius using

A = \dfrac{\pi r^2}{6}

. Give each answer correct to 2 decimal places. [3]

b

Find the percentage error between the predicted and actual area at each site using

\text{percentage error} = \frac{|\text{predicted} - \text{actual}|}{\text{actual}} \times 100\%.

Hence determine the site at which the model is least accurate, and find the difference in sod cost between the predicted and actual amounts at that site. Give the cost difference in USD, correct to 2 decimal places. [4]

c

The planner proposes ordering sod based on the model prediction for

r = 12.0

m and adding a 10% buffer for waste. Advise the planner whether this strategy is sufficient to cover the actual area at every site in the table, and identify one limitation of using this sector model for a real roundabout installation. [3]

Solutions

24QuestionAngles at the Centre and in the CircleAssessment Practice

6 marks~9 minCriterion A

The diagram shows a circle with centre

O

. Points

A

,

B

, and

C

lie on the circumference, with

C

on the major arc

AB

. The central angle

\angle AOB = 112°

.

a

Write down the value of

\angle OBA

, giving a reason. [1]

b

Find the value of

\angle OAB

. [2]

c

A fourth point

D

lies on the minor arc

AB

. The angle

\angle ADB = y°

. Justify the value of

y

, using two circle theorems. [3]

Solutions

25QuestionAngles at the Centre and in the CircleAssessment Practice

5 marks~8 minCriterion C

The diagram shows a circle with centre

O

. Points

A

,

B

,

C

and

D

lie on the circle.

C

lies on the major arc

AB

and

D

lies on the minor arc

AB

. The angle

AOB = 80°

.

a

Write down the size of angle

ACB

. [1]

b

Find the size of angle

ADB

. Show all working. [2]

c

A student claims that angles

ACB

and

ADB

are supplementary. Justify whether this claim is correct. [2]

Solutions

26QuestionAngles 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

27QuestionTangent Properties and Circle Geometry ProofsAssessment Practice

2 marks~3 minCriterion C

A telecommunications engineer is positioning a satellite dish. The signal cable runs from point

A

on the ground directly to point

B

on the outer edge of a circular support ring, meeting the ring at exactly one point.

The diagram shows a circle with centre

O

. The line

AB

is tangent to the circle at point

B

.

Justify why

OB \perp AB

, and explain how this perpendicularity guarantees the cable meets the ring at exactly one point. [2]

Solutions

28QuestionAngles 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

29QuestionRotation around a given pointAssessment Practice

7 marks~11 minCriterion B

The diagram shows triangle

ABC

with vertices

A(1, 2)

,

B(4, 2)

, and

C(4, 5)

.

Triangle

ABC

is first rotated

90^\circ

clockwise about the origin to give triangle

A_1B_1C_1

, then reflected in the line

y = -x

to give triangle

A'B'C'

.

a

Find the coordinates of

A_1

,

B_1

, and

C_1

after the

90^\circ

clockwise rotation about the origin. [2]

b

Find the coordinates of

A'

,

B'

, and

C'

after the reflection in the line

y = -x

. [2]

c

Justify that a single transformation maps triangle

ABC

directly onto triangle

A'B'C'

, stating the mirror line and verifying your answer using the coordinates of at least two vertex pairs. [3]

Solutions

30QuestionClassifying shapes and anglesAssessment Practice

6 marks~9 minCriterion C

The diagram shows triangle

ABC

with

AB = 10

cm,

BC = 7

cm, and

\angle BAC = 38^\circ

.

a

State why the given information produces an ambiguous case when applying the sine rule. [1]

b

Find the two possible values of

\angle BCA

, giving your answers to one decimal place. [2]

c

For each value of

\angle BCA

found in part (b), find the corresponding length

AC

in centimetres to three significant figures. Justify which value of

AC

is consistent with a surveyor's requirement that the triangle must have an obtuse angle at

C

. [3]

Solutions

31QuestionCoordinate geometry (distance, midpoint, gradient formulas)Assessment Practice

8 marks~12 minCriterion D

A city planner is designing a park on a coordinate grid where 1 unit represents 100 m. Entrance A is at

(3, 7)

and Entrance B is at

(15, 2)

. A straight footpath is planned directly between A and B.

The diagram shows the positions of A, B, and the fountain F at

(3, 2)

.

a

Show that the straight-line distance between A and B on the grid is exactly 13 units. [2]

b

A second route connects A to the fountain F at

(3, 2)

, then continues from F to B. The planner claims this two-segment route is longer than the direct path AB. Determine whether the planner's claim is correct, giving all distances in metres. [2]

c

The straight footpath AB is built. A surveyor measures the actual walking distance as 1 430 m. The percentage error between the planned length and the surveyed distance is given by

\text{percentage error} = \frac{|\text{planned} - \text{actual}|}{\text{actual}} \times 100\%

Calculate the percentage error to 1 decimal place. Advise the city planner whether the planned length is suitable to use for construction budgeting, justifying your answer with reference to the percentage error. [4]

Solutions

32QuestionAngles in intersecting and parallel linesAssessment Practice

6 marks~9 minCriterion A

The diagram shows two parallel lines

p

and

q

cut by a transversal

t

. Angle

a

, formed between

t

and line

p

on the right-hand side below

p

, measures

(3x + 15)°

. Angle

b

, in the alternate interior position between

t

and line

q

, measures

(5x - 27)°

. A third line

r

passes through the point where

t

meets

q

. The angle between

r

and

q

, measured on the same side as

b

, equals

\dfrac{1}{3}

of angle

a

.

a

State the geometric relationship between angles

a

and

b

, and hence form an equation in

x

. [2]

b

Find the value of

x

and calculate the size of angle

a

. [2]

c

Justify whether line

r

is perpendicular to line

t

. [2]

Solutions

33QuestionArea of Plane Figures: triangle, rectangle, square, parallelogram, rhombus, kite, trapezium, circleAssessment Practice

6 marks~9 minCriterion D

A homeowner plans to turf a rectangular garden. The measured dimensions are

8.2

m by

5.4

m, and the tape measure has a maximum absolute error of

\pm 0.05

m in each measurement.

a

Write down the upper bound and lower bound of each dimension. [1]

b

Calculate the maximum possible area and the minimum possible area of the garden. Give your answers in m², correct to 2 decimal places. [2]

c

Turf is sold in whole square metres only. The homeowner buys exactly

45

m². Advise the homeowner whether this purchase guarantees the entire garden is covered, regardless of the true dimensions, justifying your advice with reference to your answer from part (b). [3]

Solutions

34QuestionArea of Plane Figures: triangle, rectangle, square, parallelogram, rhombus, kite, trapezium, circleAssessment Practice

6 marks~9 minCriterion B

The diagram shows the first three figures of a pattern made from trapeziums. In Figure

n

, each trapezium has parallel sides of length

n

cm and

(n + 4)

cm and a perpendicular height of

3

cm. Figure

n

contains exactly

n

such trapeziums arranged in a row.

a

Write down the number of trapeziums in Figure 4. [1]

b

Find a rule for the total area

A

cm

^2

of Figure

n

, giving your answer in the form

A = an^2 + bn + c

, where

a

,

b

, and

c

are integers. [2]

c

A student claims that one of the figures in this pattern has a total area of

270

cm

^2

. Justify whether the student's claim is correct. [3]

Solutions

35QuestionArea of Plane Figures: triangle, rectangle, square, parallelogram, rhombus, kite, trapezium, circleAssessment Practice

6 marks~9 minCriterion A

The diagram shows the graph of

f(x) = x^2 + bx + c

for

-1 \le x \le 5

, where

b

and

c

are integers. The graph passes through the point

(5, 0)

and has axis of symmetry

x = 2

.

a

Write down the coordinates of the other

x

-intercept of the graph. [1]

b

Find the values of

b

and

c

. [3]

c

The horizontal line

y = k

intersects the graph of

f

at exactly one point within the domain

-1 \le x \le 5

. Justify whether there is more than one value of

k

for which this is true. [2]

Solutions

36QuestionArea of Plane Figures: triangle, rectangle, square, parallelogram, rhombus, kite, trapezium, circleAssessment Practice

7 marks~11 minCriterion C

A rectangle has a fixed perimeter of 20 cm. Let

l

represent the length (in cm) and

w

represent the width (in cm), where

l > w > 0

.

The table below shows four possible dimensions.

$l$ (cm)	9	8	7	6
$w$ (cm)	1	2	3	4
$A$ (cm²)	?	?	?	?

a

Complete the table by calculating the area for each pair of dimensions. Give each area in cm². [2]

b

A student proposes the formula

A = l(10 - l)

to model the area of any rectangle with perimeter 20 cm and length

l

.

Show that this formula is consistent with the perimeter constraint

2l + 2w = 20

, and hence find the exact value of

l

that gives the maximum area, stating the maximum area in cm². [3]

c

The student claims: "The maximum area always occurs exactly halfway between the two lengths for which the area equals zero, because the quadratic

A = l(10 - l)

is symmetric about its axis of symmetry."

Justify whether this claim is mathematically correct, and assess whether the maximum area is actually attainable given the constraint

l > w > 0

. [2]

Solutions

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