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Geometry and Measurement

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

1QuestionConstructing Perpendicular and Angle BisectorsConcept Practice
2 marks~3 minCriterion A
A landscape architect is designing a triangular garden plot ABCABC. To install a straight irrigation pipe from vertex BB to a point DD on side ACAC, the pipe must bisect ∠ABC\angle ABC exactly, so that each flower bed on either side receives equal angular coverage from the sprinkler at BB.

Deduce the relationship between ∠ABD\angle ABD and ∠CBD\angle CBD, and state the value of each angle if ∠ABC=74°\angle ABC = 74°. [2]
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2QuestionMeasuring 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°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]
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3QuestionInterior 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°360°.

Using the formula S=(n−2)×180°S = (n-2) \times 180°, where nn 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]
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4QuestionDensity 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.

Density=MassVolume\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]
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5QuestionUsing 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 △ABC≅△DEF\triangle ABC \cong \triangle DEF. In △ABC\triangle ABC, side AB=5AB = 5 cm, side BC=7BC = 7 cm, and ∠ABC=64°\angle ABC = 64°.
a
Deduce the side in △DEF\triangle DEF that corresponds to side ABAB, and state its length. [1]
b
Justify whether the engineer can conclude that ∠DEF=64°\angle DEF = 64° without taking any further measurements of △DEF\triangle DEF. [1]
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6QuestionAngles at the Centre and in the CircleConcept Practice
2 marks~3 minCriterion A
A telecommunications engineer is positioning a signal repeater at point CC on the boundary of a circular coverage zone. The transmitter and receiver are fixed at points AA and BB on the boundary, with the network hub at centre OO. The engineer knows that ∠ACB=35°\angle ACB = 35°.
a
Deduce the measure of ∠AOB\angle AOB. [1]
b
The hub OO can only relay signals when the central angle ∠AOB\angle AOB is strictly less than 75°75°. Justify whether this network configuration meets the relay condition. [1]
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7QuestionConstructing Perpendicular and Angle BisectorsAssessment Practice
6 marks~9 minCriterion B
Points A(x1,y1)A(x_1, y_1) and B(x2,y2)B(x_2, y_2) lie in the Cartesian plane. Their midpoint is MM and the perpendicular bisector of ABAB passes through MM.

Four cases are given.

A(0,0), B(2,2)A(0,0),\ B(2,2): M=(1,1)M=(1,1), perpendicular bisector y=−x+2y = -x+2

A(1,2), B(3,4)A(1,2),\ B(3,4): M=(2,3)M=(2,3), perpendicular bisector y=−x+5y = -x+5

A(−1,1), B(1,5)A(-1,1),\ B(1,5): M=(0,3)M=(0,3), perpendicular bisector y=−0.5x+3y = -0.5x+3

A(2,−1), B(4,3)A(2,-1),\ B(4,3): M=(3,1)M=(3,1), perpendicular bisector y=−0.5x+2.5y = -0.5x+2.5
a
Analyse the relationship between the slope of ABAB, the slope of its perpendicular bisector, and the coordinates of MM in each case. [2]
b
Deduce a general equation for the perpendicular bisector of ABAB in the form y=mx+cy = mx + c, where mm and cc are expressed in terms of x1, y1, x2, y2x_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)A(3,-2) and B(7,4)B(7,4). The cable must lie along the perpendicular bisector of ABAB. Apply your formula from part (b) to find the equation of this line, then justify whether the point (5,1)(5,1) lies on the cable's path. [1]

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8QuestionConstructing Perpendicular and Angle BisectorsAssessment Practice
6 marks~9 minCriterion C
Landscape designer Zara is planning a garden path that must be equidistant from two fixed points, AA and BB, marked on a scale drawing. The line segment ABAB is drawn below.
a
Construct the perpendicular bisector of ABAB using a compass and straightedge only. All construction arcs must be visible. [2]
b
Label a point PP on the perpendicular bisector. Measure and record the distances PAPA and PBPB in centimetres. [1]
c
Calculate the percentage difference between PAPA and PBPB using:

Percentage Difference=∣PA−PB∣PA+PB2×100\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 AA and BB 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]
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9QuestionConstructing Perpendicular and Angle BisectorsAssessment Practice
4 marks~6 minCriterion D
A farmer needs to divide an irregularly shaped, hilly field equally between two children. She marks two boundary points, AA and BB, on opposite edges of the field, then constructs the perpendicular bisector of segment ABAB as the proposed fence line.
a
Explain why the perpendicular bisector of ABAB passes through the midpoint of the segment and is used as a symmetric fence line. [1]
b
The farmer uses a rope stretched between AA and BB to represent the segment. Explain two ways this method could introduce error into the construction. [1]
c
The field slopes significantly on one side. Advise the farmer whether the perpendicular bisector method alone is sufficient to guarantee an equal division of the field's actual surface area, justifying your advice. [2]
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10QuestionCreating 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 denominator5 00010 00020 00040 000
Map distance (cm)201052.5


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]

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11QuestionUnderstanding 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°030° and Station B detects the hiker on a bearing of 330°330°.
a
Deduce the size of each interior angle in triangle ABHABH, where HH 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]
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12QuestionCreating 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:n1: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]
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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 nn-sided polygon is given by

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

and the sum of interior angles by S=180(n−2)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 EE of a regular nn-sided polygon in terms of nn, 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]

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14QuestionParallel 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°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]
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15QuestionInterior and Exterior Angles of PolygonsAssessment Practice
4 marks~6 minCriterion D
A construction company is designing a floor using a tessellating pattern of regular hexagons and equilateral triangles, where exactly two hexagons and two triangles meet at every vertex.
a
Calculate the interior angle of a regular hexagon and the interior angle of an equilateral triangle. [2]
b
Show that two regular hexagons and two equilateral triangles can meet at a vertex without gaps or overlaps. [1]
c
The site engineer claims this mathematical model "guarantees a perfect fit on any real floor." Advise the engineer whether this claim should be used to guide the physical installation, referring to at least two practical factors. [1]
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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 mm= 1 \ \text{mm}
Square B: side =1 cm= 1 \ \text{cm}
Square C: side =1 dm= 1 \ \text{dm}
a
Calculate the area of each square in mm2\text{mm}^2, cm2\text{cm}^2, and m2\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]

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

Mass MM (g): 2, 8, 18, 32, 50
a
Analyse the mass values to identify the relationship between MM and ss. 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 DD (g/cm³) of a cube in terms of its side length ss, using D=MVD = \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]

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18QuestionConverting Between Metric UnitsAssessment Practice
4 marks~6 minCriterion D
A city planner collects rainfall data in litres (L) to estimate the volume of water entering a reservoir. The estimated volume is then converted to cubic metres (m3\text{m}^3) to assess the reservoir's capacity, using 1 m3=1000 L1 \, \text{m}^3 = 1000 \, \text{L}.
a
Explain how to convert a volume given in litres to cubic metres, using the conversion factor above. [1]
b
Explain one advantage of using rainfall data to estimate reservoir inflow for water resource management. [1]
c
The city planner uses only rainfall data to predict the actual volume of water stored in the reservoir. Critique this approach by identifying two distinct limitations, explaining how each causes the predicted volume to differ from the actual volume. [2]
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19QuestionCriteria for Similar TrianglesAssessment Practice
6 marks~9 minCriterion D
Surveyors use similar triangles to estimate the heights of tall structures. A surveyor stands on flat ground and measures the angle of elevation to the top of a 3-metre pole as 30°30°, and the angle of elevation to the top of a skyscraper as 60°60°. The distance from the surveyor to the base of the pole is 10 metres.
a
Explain how the AA similarity criterion establishes that the two triangles formed are similar, and use this to calculate the height of the skyscraper. [2]
b
Analyse two sources of error in this method — measurement inaccuracies and terrain irregularities — and explain how each affects the calculated height. [2]
c
Justify whether the similar-triangles method is sufficiently reliable for this surveying task, or whether additional measurements are required before the result can be used with confidence. [2]
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20QuestionUsing 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)246
Perimeter of base (cm)102030
Area of base (cm²)6.8827.5261.92
Volume (cm³)34.40275.20928.80


In each prism, the height equals the side length.
a
Deduce a formula for the perimeter PP of the pentagonal base in terms of its side length ss. [1]
b
Deduce a formula for the area AA of the pentagonal base in terms of ss, showing that the ratio As2\dfrac{A}{s^2} is constant. [2]
c
Using your results from (a) and (b), deduce a formula for the volume VV of the prism in terms of ss. [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]
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21QuestionCriteria for Triangle Congruence SSS SAS ASA RHSAssessment Practice
6 marks~9 minCriterion C
A structural engineer is checking whether two triangular roof trusses, △ABC\triangle ABC and △DEF\triangle DEF, are identical in shape and size before installation. She records measurements progressively.
a
She finds AB=DEAB = DE and BC=EFBC = EF. Explain why this information alone is insufficient to confirm the trusses are congruent. [2]
b
She then confirms ∠ABC=∠DEF\angle ABC = \angle DEF. Justify why △ABC≅△DEF\triangle ABC \cong \triangle DEF can now be concluded, naming the congruence criterion used. [2]
c
A colleague suggests that knowing only ∠ABC=∠DEF\angle ABC = \angle DEF and BC=EFBC = EF (without AB=DEAB = 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]
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22QuestionLength of arc and chord, perimeter and area of sector and segmentAssessment Practice
6 marks~9 minCriterion D
A pizza restaurant advertises that its Large pizza has a diameter of 40 cm and is cut into 8 equal slices. The restaurant claims each slice has a crust length (arc length) of 16 cm.

The diagram shows one slice of the pizza, with centre OO, radius rr, and the arc length of one slice labelled.
a
Calculate the actual arc length of one slice. Give your answer in exact form and to 3 significant figures. [2]
b
The restaurant's manager states: "The percentage error in our advertised crust length is less than 10%."

The percentage error is calculated using:

percentage error=∣claimed−actual∣actual×100%\text{percentage error} = \frac{|\text{claimed} - \text{actual}|}{\text{actual}} \times 100\%

Determine whether the manager's statement is correct. Show all working and give your percentage error to 1 decimal place. [2]
c
The restaurant now considers cutting the same Large pizza into nn equal slices so that each slice has an arc length of exactly 5π2\dfrac{5\pi}{2} cm.

Find the value of nn and justify whether this cutting is physically possible for the restaurant. [2]
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23QuestionLength 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)306090120
Arc length (cm)5.2410.4715.7120.94
Sector area (cm²)26.1852.3678.54104.72
a
Describe the relationship between the central angle θ\theta and the arc length LL, 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 LL of a sector with central angle θ\theta degrees and radius rr cm. Hence find the arc length of a sector with central angle 150∘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 rr cm. Two sectors of this circle have central angles α\alpha degrees and β\beta degrees respectively, where α+β=180\alpha + \beta = 180. Show that the sum of their arc lengths equals πr\pi r, and justify why this result holds regardless of the individual values of α\alpha and β\beta. [3]
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24QuestionLength of arc and chord, perimeter and area of sector and segmentAssessment Practice
8 marks~12 minCriterion C
A circle has centre OO and radius r=10r = 10 cm. Points AA and BB lie on the circumference. The shaded region is the minor segment cut off by chord ABAB, where the central angle ∠AOB=θ\angle AOB = \theta degrees. The area of the minor segment is given by

A(θ)=θ360×πr2−12r2sin⁡θA(\theta) = \frac{\theta}{360} \times \pi r^2 - \frac{1}{2}r^2 \sin\theta

where r=10r = 10 cm.

θ\theta (degrees): 60, 90, 120
a
Calculate the area of the minor segment for each value of θ\theta. Give each answer correct to 1 decimal place. [3]
b
A student claims: "The segment area increases by the same amount each time θ\theta increases by 30°30°." Using your values from part (a), determine whether this claim is correct. Show all working. [2]
c
The formula contains a sector-area term θ360πr2\dfrac{\theta}{360}\pi r^2 and a triangle-area term 12r2sin⁡θ\dfrac{1}{2}r^2\sin\theta. Justify why the segment area increases as θ\theta increases from 60°60° to 120°120°, by analysing the behaviour of each term separately over this interval. [3]
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25QuestionLength of arc and chord, perimeter and area of sector and segmentAssessment Practice
5 marks~8 minCriterion A
The diagram shows a sector with radius r=10r = 10 cm and central angle θ=72°\theta = 72°.
a
Write down the arc length of this sector, giving your answer in the form kπk\pi cm, where kk is an integer. [1]
b
A second sector has the same central angle of 72°72° and an arc length of 10π10\pi cm. Find the radius RR of the second sector. Give your answer in centimetres. [2]
c
A student claims: "If the central angle stays the same, doubling the arc length always doubles the radius." Justify whether this claim is correct, using the arc length formula and your results from parts (a) and (b). [2]
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26QuestionAngles in the Same Segment and Alternate Segment TheoremAssessment Practice
5 marks~8 minCriterion C
Points AA, BB, CC, DD, and EE lie on the circumference of a circle with centre OO. The line segment AEAE is a diameter. Angle BAC=34°BAC = 34°. A surveying team uses this circular arrangement to model sight-lines between observation posts, where a 90°90° angle between lines confirms a perpendicular alignment required for accurate triangulation.
a
Deduce the measure of angle BCEBCE. [1]
b
Calculate angle BECBEC and justify which circle theorem you used. [2]
c
Angle EDCEDC is claimed to equal angle BACBAC. Justify whether this claim is correct, and explain what this means for the sight-lines from posts DD and AA to the arc ECEC. [2]
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27QuestionAngles in the Same Segment and Alternate Segment TheoremAssessment Practice
6 marks~9 minCriterion D
A ship's captain navigates between two coastal landmarks, AA and BB, avoiding a submerged reef. By measuring the angle θ\theta subtended by chord ABAB from the ship's position, the captain uses the angles in the same segment theorem to stay on a safe circular arc that keeps the reef outside it.
a
Explain how the angles in the same segment theorem guarantees that a constant angle θ\theta keeps the ship on the same circular arc through AA and BB. [2]
b
Explain two limitations of using angle measurements alone to maintain this safe arc in practice. [2]
c
Advise the captain whether this geometric method alone is sufficient for safe navigation, and justify your advice by recommending at least two specific additional tools or strategies that should be used. [2]
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28QuestionAngles in the Same Segment and Alternate Segment TheoremAssessment Practice
5 marks~8 minCriterion B
A satellite dish is mounted on a curved support. An engineer models a cross-section of the support as a circle with centre OO. A tangent PTPT touches the circle at point AA, and point BB lies on the circumference. The angle BAP=θBAP = \theta.
a
State the size of angle OAPOAP, justifying your answer with the appropriate theorem. [1]
b
Prove that triangle OABOAB is isosceles and hence deduce angle OABOAB in terms of θ\theta. [2]
c
Prove that ∠AOB=2θ\angle AOB = 2\theta, then advise the engineer whether measuring the tangent angle θ\theta at the mounting point AA is a sufficient method for determining the central angle of the support. [2]
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29QuestionCoordinate geometry (distance, midpoint, gradient formulas)Assessment Practice
8 marks~12 minCriterion C
The diagram shows four points A(1, 2)A(1,\, 2), B(3, 6)B(3,\, 6), C(6, 12)C(6,\, 12), and D(4, 9)D(4,\, 9) plotted on a coordinate grid.
a
Show that AA, BB, and CC are collinear by calculating the gradients of ABAB and ACAC, and stating what these values imply. [3]
b
A fifth point P(k, 4k−4)P(k,\, 4k-4) lies on the line through AA, BB, and CC. Find the value of kk. [2]
c
A student claims that because D(4, 9)D(4,\, 9) is close to line ACAC, the gradient condition for collinearity is "approximately satisfied", so AA, BB, CC, DD are "approximately collinear". Justify why this reasoning is mathematically invalid, referring to the exact gradient values in your answer. [3]
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30QuestionSymmetry and reflectionAssessment Practice
6 marks~9 minCriterion D
A graphic designer is creating a logo for a café. The left half of the logo is a quadrilateral with vertices A(1,2)A(1, 2), B(2,4)B(2, 4), C(3,2)C(3, 2), D(2,0)D(2, 0), joined in order. The designer reflects this quadrilateral over the line x=5x = 5 to produce the right half, with images A′A', B′B', C′C', D′D'.

The diagram shows the left half plotted on a coordinate grid.
a
State the coordinates of A′A', B′B', C′C', and D′D' after reflection over the line x=5x = 5. [1]
b
The designer makes a consistent measurement error: every reflected point is placed 0.30.3 units too far to the right of its correct position. Write down the erroneous coordinates of A′A', B′B', C′C', D′D', and calculate the percentage error in the xx-coordinate of B′B', giving your answer to one decimal place.

percentage error=∣measured−correct∣∣correct∣×100%\text{percentage error} = \frac{|\text{measured} - \text{correct}|}{|\text{correct}|} \times 100\% [3]
c
The designer claims the 0.30.3-unit error is small enough that the completed logo still looks symmetric about x=5x = 5. Assess this claim, and suggest one limitation of relying on a coordinate grid with 11-unit spacing to detect this type of error in a real logo design. [2]
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31QuestionMovement on a plane – isometric transformations, enlargements, tessellationsAssessment Practice
6 marks~9 minCriterion B
The diagram shows the first three figures of a pattern made from unit squares arranged in an L-shape.

Figure 1 has 3 unit squares, Figure 2 has 7 unit squares, and Figure 3 has 13 unit squares.
a
Write down the number of unit squares in Figure 4. [1]
b
Find a rule for the number of unit squares SS in Figure nn, giving your answer in the form S=an2+bn+cS = an^2 + bn + c, where aa, bb and cc are integers. Show clearly how you obtained your values of aa, bb and cc. [3]
c
Show that the total number of unit squares across Figure nn and Figure n+1n+1 can be written as 2(n2+2n+2)2(n^2 + 2n + 2). Hence justify why this total is always even for every positive integer nn. [2]
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32QuestionMovement on a plane – isometric transformations, enlargements, tessellationsAssessment Practice
6 marks~9 minCriterion A
The diagram shows the first four figures of a pattern. Each figure is a triangle with vertices A(n)A^{(n)}, B(n)B^{(n)}, C(n)C^{(n)}. Figure 1 has A(1)=(1,1)A^{(1)} = (1, 1), B(1)=(3,1)B^{(1)} = (3, 1), C(1)=(2,4)C^{(1)} = (2, 4). Each new figure is obtained by translating the previous triangle 3 units in the xx-direction and 2 units in the yy-direction.
a
Write down the coordinates of A(3)A^{(3)}. [1]
b
Find a general rule for the coordinates of A(n)A^{(n)}, where nn is a positive integer. [2]
c
A designer claims that Figure 21 is the first figure for which both coordinates of A(n)A^{(n)} exceed 40. Justify whether this claim is correct. [3]
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33QuestionArea of Plane Figures: triangle, rectangle, square, parallelogram, rhombus, kite, trapezium, circleAssessment Practice
8 marks~12 minCriterion B
The diagram shows Figures 1, 2, and 3 of a sequence of L-shaped figures made from unit squares. Figure 1 contains 3 unit squares, Figure 2 contains 10 unit squares, and Figure 3 contains 21 unit squares.
a
Write down the number of unit squares in Figure 4. [1]
b
Find a formula for the number of unit squares SS in Figure nn, in the form S=an2+bnS = an^2 + bn, where aa and bb are integers. Show all working. [3]
c
Justify why the formula must contain an n2n^2 term by referring to the structure of the L-shaped figure, and hence deduce the smallest value of nn for which S>200S > 200. [4]
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34QuestionCompound Shapes: mixed figures, real-world problemsAssessment Practice
8 marks~12 minCriterion A
A landscape architect designs a garden path by joining a rectangular section to a square section, forming an L-shaped compound region, as shown in the diagram. The rectangle has length 99 m and width 44 m. The square has side length 33 m.

(a) Show that the total area of the L-shaped path is 4545 m². [2]

(b) A uniform layer of gravel covers the entire path to a depth of 0.150.15 m. The cost of gravel is 2828 USD per m³. Find the total cost of gravel required. Give your answer in USD, correct to the nearest dollar. [2]

(c) The architect proposes enlarging the entire L-shaped path using a linear scale factor kk, where kk is a positive integer greater than 11, so that the new total area is at least 405405 m².

(i) Find the smallest integer value of kk that satisfies this requirement. [2]

(ii) The gravel depth remains 0.150.15 m and the cost per m³ remains 2828 USD. The project budget for gravel is 18001800 USD. Evaluate whether the total cost of gravel for the enlarged path, when kk takes the value found in part (c)(i), is affordable within this budget, justifying your answer with supporting calculations. [2]
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35QuestionArea of Plane Figures: triangle, rectangle, square, parallelogram, rhombus, kite, trapezium, circleAssessment Practice
13 marks~20 minCriterion D
A homeowner is renovating an L-shaped room and needs to order hardwood flooring. The floor plan is divided into a rectangle and a trapezium. The rectangle has dimensions 66 m by 44 m. The trapezium has parallel sides of 44 m and 22 m, and a perpendicular height of 33 m.

The homeowner measures each wall to the nearest 55 cm.
a
Calculate the total area of the L-shaped floor. Give your answer in m². [3]
b
Determine the maximum possible total floor area, given the measurement tolerances stated above. Give your answer in m² correct to 3 significant figures. [4]
c
The flooring is sold in whole packs, each covering exactly 55 m². The homeowner orders the minimum number of whole packs needed to cover the nominal area from part (a).

Advise the homeowner whether this order is sufficient to cover the maximum possible floor area from part (b), justifying your advice with a calculation. [3]
d
Suggest one limitation of modelling the floor as a rectangle and a trapezium, and explain how this limitation could cause the ordered flooring to be insufficient even if the maximum area from part (b) is fully covered. [3]
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36QuestionVolume and Surface Area: cube, cuboid, cylinderAssessment Practice
8 marks~12 minCriterion C
A storage company designs open-top rectangular boxes (no lid). One box has base length l=8l = 8 cm, base width w=3w = 3 cm, and height h=4h = 4 cm.
a
Calculate the total surface area of the open-top box, showing the area of each pair of faces separately. Give your answer in cm². [3]
b
The company scales up the box using an enlargement with linear scale factor k=1.5k = 1.5. Find the total surface area of the enlarged box. Give your answer in cm². [2]
c
A client claims that scaling a box by linear scale factor kk multiplies its surface area by kk. Using your answers to parts (a) and (b), critique this claim and state the correct relationship between the surface area of the original box and the surface area of the enlarged box. [3]
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