Similarity

The diagram shows two triangles, ABC and PQR, and it is marked NOT TO SCALE. In triangle ABC, AC measures 1.61 cm and CB measures 2.8 cm. In triangle PQR, RQ measures 3.2 cm and PR is labelled $x$ cm. Triangle ABC is mathematically similar to triangle PQR.

Feb/March 2021

Determine the value of $x$.

Feb/March 2021

From the diagram, AB and CD are parallel, and CB together with AD meet at X. AB = 3.0 cm, AX = 2.0 cm, BX = 2.7 cm and CD = 7.5 cm.

Feb/March 2023

The two solids are mathematically similar, and their volumes are 81 cm$^3$ and 24 cm$^3$. The smaller solid has surface area 44 cm$^2$.

Feb/March 2023

Triangle $ABC$ is similar, in the mathematical sense, to triangle $DEF$. For triangle $ABC$, $AC = 118.9$ cm and $AB = h$ cm. For triangle $DEF$, $DF = 159.9$ cm and $DE = 97.5$ cm. The diagram is labelled NOT TO SCALE.

Feb/March 2024

Triangles $ABC$ and $PQR$ are mathematically similar.

Feb/March 2025

Triangles $ABC$ and $DEF$ are similar. For triangle $ABC$, $AB = 8\,\text{cm}$, $BC = 10\,\text{cm}$ and $AC = y\,\text{cm}$. For triangle $DEF$, $DE = 6\,\text{cm}$, $EF = x\,\text{cm}$ and $DF = 9\,\text{cm}$. The diagrams are labelled NOT TO SCALE.

May/June 2015

Triangle ABC and triangle DEF are similar. Diagram information: For triangle ABC, AB = 8 cm, BC = 10 cm, and AC = $y$ cm. For triangle DEF, DE = 6 cm, EF = $x$ cm, and DF = 9 cm. Diagram is NOT TO SCALE.

May/June 2015

Calculate the angle $EFA$.

May/June 2015

Calculate the height of the water in the tank.

May/June 2015

Triangles $ABC$ and $DEF$ are similar. In triangle $ABC$, $AC = 6\,$cm and $BC = 8.4\,$cm. In triangle $DEF$, $DF = 15\,$cm.

May/June 2016

Triangles $A$ and $B$ are similar. For triangle $A$, one side measures 12 cm, and the matching side in triangle $B$ measures 20 cm. Another side of triangle $A$ measures 9 cm, and the matching side in triangle $B$ is $x$ cm. The diagram is not drawn to scale.

May/June 2016

Determine the value of $k$.

May/June 2016

$ABC$ and $DEF$ are similar triangles. For triangle $ABC$, $AB = 15$ cm and the base $BC = 16.5$ cm. In triangle $DEF$, $DE = 5$ cm. The diagrams are NOT TO SCALE.

May/June 2017

The two barrels in the diagram are mathematically similar. The smaller barrel stands at a height of $h$ cm and has a capacity of 100 litres. The larger barrel measures 90 cm in height and has a capacity of 160 litres.

May/June 2017

Within the diagram, $AB$ and $CD$ run parallel to each other. $AD$ and $BC$ meet at $X$. $AB = 8\text{ cm}$, $CD = 4\text{ cm}$, $CX = 2\text{ cm}$ and $DX = 2.8\text{ cm}$. The figure is labelled NOT TO SCALE.

May/June 2017

The two bottles, along with their labels, are mathematically similar. The smaller bottle holds 0.512 litres of water and has a label with area $96\text{ cm}^{2}$. The larger bottle holds 1 litre of water.

May/June 2017

Calculate the length of $PB$.

May/June 2018

Two triangles are illustrated. In triangle $ABC$, the angles measure $65^\circ$, $75^\circ$, and $40^\circ$, and the side $AB = 6$ cm. In triangle $PQR$, the matching angles are $65^\circ$, $75^\circ$, and $40^\circ$, with base $PR = 27$ cm and side $PQ = 18$ cm. NOT TO SCALE.

May/June 2019

The shapes ABCD and PQRS are displayed. The marked lengths are 5.5 cm, $a$ cm, 7.7 cm and 9.8 cm, and the diagram is NOT TO SCALE.

May/June 2019

The diagram depicts two regular pentagons. Pentagon $FGHJK$ is an enlargement of pentagon $ABCDE$, with centre $O$. The diagram is not drawn to scale.

May/June 2019

The diagram displays two cylinders. The smaller cylinder is 5 cm in diameter and 13 cm high. The larger cylinder is 12.5 cm in diameter and 32.5 cm high. The diagram is not drawn to scale.

May/June 2021

The diagram shows two triangles. Triangle $ABC$ has sides $AB$, $BC = 34\text{ cm}$ and $AC = 28\text{ cm}$. Triangle $DEF$ has sides $DE$, $EF = 85\text{ cm}$ and $DF$. The diagram is not drawn to scale. Triangle $ABC$ is similar to triangle $DEF$.

May/June 2021

Complete the table that shows whether each pair of triangles is congruent. The first two rows have already been completed for you. None of the diagrams is drawn to scale.

May/June 2021

In the diagram, $AB$ runs parallel to $CD$. $AD$ and $BC$ cross at $X$. $AB = 6\text{ cm}$, $CD = 12\text{ cm}$, $CX = 8\text{ cm}$ and $DX = 7\text{ cm}$.

May/June 2021

Triangle $ABC$ and triangle $PQR$ are mathematically similar.

May/June 2022

Triangle $ABC$ is similar to triangle $PQR$ in the mathematical sense. Diagram labels: For triangle $ABC$, $AB = 8\text{ cm}$ and $BC = 6\text{ cm}$. For triangle $PQR$, $PQ = 6\text{ cm}$. Diagram marked NOT TO SCALE.

May/June 2022

Explain, with reasons, why triangle $PMR$ is similar to triangle $MSR$.

May/June 2022

Triangles $ABC$ and $DEF$ are similar. For triangle $ABC$, $BC = 7.2$ cm and $AC = 5.6$ cm. For triangle $DEF$, $EF = 8.1$ cm and $DF = h$ cm. The diagrams are marked NOT TO SCALE.

May/June 2023

The figure shows two triangles. For triangle ABC, AB = 15 cm and BC = 9 cm. For triangle PQR, PQ = 18 cm. The diagram is NOT TO SCALE. Triangle ABC is similar to triangle PQR.

May/June 2023

Two right-angled triangles are illustrated. In triangle $ABC$, the base is $BC = 7.2\text{ cm}$ and the height is $AC = 5.6\text{ cm}$. In triangle $DEF$, the base is $EF = 8.1\text{ cm}$ and the height is $DF = h\text{ cm}$. The diagram is labelled NOT TO SCALE.

May/June 2023

Calculate the smaller parcel’s height.

May/June 2024

Rectangle A is similar, in the mathematical sense, to rectangle B.

May/June 2025

Solid A and solid B are mathematically similar. Solid A has height 7 cm and surface area 60 cm$^2$, whereas the surface area of solid B is 540 cm$^2$. A diagram of both solids is shown, not drawn to scale.

May/June 2025

There are two triangles in the diagram. Triangle $ABC$ has base $AC = 12\text{ cm}$. Triangle $UVW$ has base $UW = 21\text{ cm}$ and side $UV = 14\text{ cm}$. Triangle $ABC$ is mathematically similar to triangle $UVW$. The diagram is labelled NOT TO SCALE.

May/June 2025

Triangles ABC and PQR are mathematically similar. The sides of triangle ABC measure 9 cm, 12 cm and 16 cm. A matching side on triangle PQR is 6 cm long. The diagrams are not drawn to scale.

May/June 2025

The diagram presents two jugs that are mathematically similar. The shorter jug measures $15\text{ cm}$ in height and $7.2\text{ cm}$ across the base. The taller jug measures $25\text{ cm}$ in height and has base width $x\text{ cm}$. The diagram is marked NOT TO SCALE.

Oct/Nov 2015

The figure displays two triangles, $ABC$ and $PQR$, that are similar. In triangle $ABC$, $AB = 10$ cm, $BC = 6$ cm and $AC = x$ cm. In triangle $PQR$, $PQ = 6.25$ cm, $PR = 7.5$ cm and $QR = y$ cm. The figure is labelled NOT TO SCALE.

Oct/Nov 2015

(a) The diagram depicts two jugs that are mathematically similar. The shorter jug has height $15\text{ cm}$ and base width $7.2\text{ cm}$. The taller jug has height $25\text{ cm}$ and base width $x\text{ cm}$. (b) The diagram depicts two glasses that are mathematically similar. The larger glass has height $16\text{ cm}$ and volume $375\text{ cm}^3$. The smaller glass has height $y\text{ cm}$ and volume $192\text{ cm}^3$.

Oct/Nov 2015

The two containers are mathematically similar. Their volumes are $54\text{ cm}^3$ and $128\text{ cm}^3$. The smaller container has a height of $4.5\text{ cm}$.

Oct/Nov 2015

Four triangles, $A$, $B$, $C$ and $D$, are illustrated with angle markings: triangle $A$ has angles $55^\circ$ and $75^\circ$; triangle $B$ has angles $55^\circ$ and $80^\circ$; triangle $C$ has angles $35^\circ$ and $80^\circ$; triangle $D$ has angles $45^\circ$ and $55^\circ$. The diagrams are marked "NOT TO SCALE".

Oct/Nov 2016

Triangles $ABC$ and $DEF$ are similar. In triangle $ABC$, one side measures 16 cm and another measures 12 cm. In triangle $DEF$, the matching sides are $x$ cm and 30 cm. The diagram is not drawn to scale.

Oct/Nov 2016

The two cups are mathematically similar. The bigger cup has a capacity of $0.5$ litres and a height of $8$ cm, whereas the smaller cup has a capacity of $0.25$ litres.

Oct/Nov 2016

A backpack with a capacity of $30$ litres has a length of $53\text{ cm}$.

Oct/Nov 2016

The diagram shows two shapes. For the first shape, the base is $15\,\text{cm}$, the upper side is $12\,\text{cm}$, and the sloping left side is marked $x\,\text{cm}$. For the second shape, which is similar to the first, the base is $12.5\,\text{cm}$, the sloping left side is $5\,\text{cm}$, and the upper side is marked $y\,\text{cm}$. The diagram is not drawn to scale.

Oct/Nov 2017

Triangle $ABC$ and triangle $PQR$ are similar. In triangle $ABC$, $AB = 8\text{ cm}$ and $BC = 12\text{ cm}$. In triangle $PQR$, $PQ = 14\text{ cm}$ and $QR = x\text{ cm}$. The diagram is not drawn to scale.

Oct/Nov 2018

Triangle $ABC$ is similar to triangle $DEF$.

Oct/Nov 2018

The diagram shows two triangles. In triangle $ABC$, the base has length $BC = 8$ cm and the height is marked $x$ cm. In triangle $PQR$, the base is $QR = 20$ cm and one side is marked $23$ cm. The diagrams are not drawn to scale.

Oct/Nov 2018

The diagram shows AB and CD as parallel lines. AD and BC meet at X at right angles. AB is 10 cm, CD is 5 cm, AX is 8 cm and BX is 6 cm.

Oct/Nov 2018

The grid contains several quadrilaterals, each labelled A, B, C, D, E, F, G and H.

Oct/Nov 2019

Two containers that are mathematically similar have heights of 30 cm and 75 cm. The larger one has a capacity of 5.5 litres.

Oct/Nov 2019

The diagram depicts two similar triangles $TUV$ and $XYZ$. In triangle $TUV$, the side matching $XZ$ is 12 cm long. In triangle $XYZ$, $YX = 63$ cm and $XZ = 54$ cm. The diagram is labelled NOT TO SCALE.

Oct/Nov 2020

A paperweight is 4 cm tall and has a volume of $38.4\,\text{cm}^3$. A paperweight that is mathematically similar has a height of 7 cm.

Oct/Nov 2020

The diagram shows three mathematically similar shapes. Their heights are in the ratio small : medium : large = $1 : 5 : 8$. The diagram is not drawn to scale.

Oct/Nov 2021

Triangles $ABC$ and $DEF$ are mathematically similar. In triangle $ABC$, $AB = 51\text{ cm}$. In triangle $DEF$, $DE = 6\text{ cm}$ and the base $EF = 4\text{ cm}$. The sketch is not drawn to scale.

Oct/Nov 2022

Triangle $PQR$ is similar to triangle $ABC$, with $\frac{PR}{AC} = \frac{2}{3}$. Also, $AB = 9\text{ cm}$ and the area of triangle $ABC$ is $18\text{ cm}^2$. Diagrams are not drawn to scale.

Oct/Nov 2022

The volumes of two mathematically similar objects are $56\text{ cm}^3$ and $875\text{ cm}^3$. The smaller object has a height of 18 cm.

Oct/Nov 2022

The two bottles are mathematically similar. The smaller bottle has a capacity of 324 ml and measures 12 cm in height. The larger bottle has a capacity of 768 ml.

Oct/Nov 2022

Triangle $ABC$ is similar to triangle $DEF$ mathematically. The measurements are $AC = 5.5$ cm, $BC = 7.3$ cm, $DF = 4.4$ cm and $EF = x$ cm. The diagram is not drawn to scale.

Oct/Nov 2023

The diagram shows two shapes that are mathematically similar. The larger shape measures $9\text{ cm}$ in width and $6.3\text{ cm}$ in height. The smaller shape measures $5\text{ cm}$ in width and $h\text{ cm}$ in height. The diagrams are marked NOT TO SCALE.

Oct/Nov 2023

Shapes A and B are mathematically similar. In shape A, the base length is 6.75 cm and the matching height is $x$ cm. In shape B, the base length is 2.7 cm and the matching height is 3.2 cm.

Oct/Nov 2024

Two solid steel statues are mathematically similar. The smaller statue is 12 cm tall and the larger statue is 15 cm tall. The mass of the larger statue is 2.5 kg. The density of steel is $8\,\text{g cm}^{-3}$. Calculate the volume of the smaller statue. [Density = mass \div volume.]

Oct/Nov 2024

The large box and the small box are mathematically similar. The volume of the large box is $72.8\%$ greater than the volume of the small box. The small box has length $3.5\,\text{cm}$ and the large box has length $x\,\text{cm}$. Diagrams of the small and large boxes are shown. NOT TO SCALE.

Oct/Nov 2024

The volumes of two mathematically similar solids are $81\text{ cm}^3$ and $24\text{ cm}^3$. The smaller solid has a height of $4.8$ cm.

Oct/Nov 2024

The diagram shows S on PQ and T on PR. ST runs parallel to QR.

Oct/Nov 2025

Triangles ABC and PQR are similar. AC = 12 cm, PR = 28 cm, PQ = 42 cm.

Oct/Nov 2025

The figure depicts two similar triangles, A and B.

Oct/Nov 2025