Two bottles are mathematically similar. The larger has capacity 1.35 litres and height 24 cm. The smaller has capacity 0.4 litres.

Calculate the smaller height. [3]

Two solids are similar. The larger has volume \(64\text{ cm}^3\) and height \(10\text{ cm}\); the smaller has volume \(27\text{ cm}^3\) and height \(h\text{ cm}\). Find \(h\). [3]

Two bottles are similar. The larger has volume \(216\text{ cm}^3\) and height \(10\text{ cm}\). The smaller has volume \(27\text{ cm}^3\) and height \(h\text{ cm}\). Find \(h\). [3]

Cylinders A and B are similar. Their surface areas are \(80\text{ cm}^2\) and \(180\text{ cm}^2\). A has height h and B has height \(12\text{ cm}\). Find h. [3]

AB and CD are parallel. Straight lines AD and BC meet at X. \(\angle ABX=y^\circ\).

(a) Give the geometrical reason why \(\angle DCX=y^\circ\). [1]

Triangles ABX and DCX are similar. \(CX=5\text{ cm},XD=8\text{ cm},AD=20\text{ cm}\).

(b(i)) Find BX. [2]

(b(ii)) \(\angle CXD=80.5^\circ\). Find the area of CXD. [2]

Similar solid A has height 7 cm and surface area 60 cm². Solid B has surface area 540 cm². Find its height. [3]

A warehouse has floor area \(800\text{ m}^2\). Its plan has scale 1:50. Find the plan area in square centimetres. [3]

(a) Find PR. [2]

(b) The triangles are cross-sections of two similar prisms. The smaller prism has volume 1120 cm³. Find the larger volume. [2]

S lies on PQ, T lies on PR, and ST is parallel to QR.

(a) Explain why triangles PQR and PST are similar, giving geometrical reasons. [3]

(b) ST=3 cm, QR=9 cm and PS=5 cm. Find PQ. [2]

(c) Triangle PST has area 2k cm². Find the area of QRTS in terms of k. [2]

(a) The triangles are mathematically similar. Find x. [2]

(b) Similar containers have surface areas 124 cm² and 279 cm². The smaller capacity is 56 ml. Find the larger capacity. [3]

A, B, C and D lie on a circle. ADX and BCX are straight lines. Angle BAD=x°, angle DCX=y°.

(a) Explain why x=y, giving a geometrical reason for each step. [2]

(b) Show that triangles ABX and CDX are similar. [2]

AD=15 cm, DX=9 cm, CX=12 cm.

(c(i)) Find BC. [3]

(c(ii)) Find the ratio area ABX : area CDX in the form n:1. [1]

Two parcels are similar. The larger has volume 80 cm³ and height 5.2 cm. The smaller has volume 33.75 cm³. Find its height. [3]

Two similar solids have volumes 81 cm³ and 24 cm³. The smaller is 4.8 cm tall. Find the height of the larger. [3]

Two solid steel statues are similar. Their heights are 12 cm and 15 cm. The larger has mass 2.5 kg; steel has density 8 g/cm³. Find the volume of the smaller statue. Density = mass ÷ volume. [4]

The large box has 72.8% more volume than the similar small box. Their corresponding lengths are x cm and 3.5 cm. Find x. [3]

\(AB\parallel CD\). Lines \(CB\) and \(AD\) intersect at \(X\). \(AB=3.0\text{ cm},\ AX=2.0\text{ cm},\ BX=2.7\text{ cm},\ CD=7.5\text{ cm}\).

Find the length \(BC\). [3]

Two mathematically similar solids have volumes \(81\text{ cm}^3\) and \(24\text{ cm}^3\). The smaller solid has surface area \(44\text{ cm}^2\).

Find the surface area of the larger solid. [3]

Triangle \(ABC\) is similar to triangle \(DEF\). \(BC=7.2\text{ cm},\ AC=5.6\text{ cm},\ EF=8.1\text{ cm},\ DF=h\text{ cm}\).

Calculate \(h\). [2]

\(\angle A=90^\circ, AC=(x+3)\text{ cm}, AB=(2x+5)\text{ cm}\).

(a(i)) The area is \(60\text{ cm}^2\). Show that \(2x^2+11x-105=0\). [3]

(a(ii)) Solve \(2x^2+11x-105=0\) by factorisation. [3]

(a(iii)) Calculate \(\angle ACB\). [3]

(b(i)) Triangle \(DEF\) is similar to \(ABC\) and has area \(93.75\text{ cm}^2\). Find its smallest angle. [1]

(b(ii)) Find the shortest side of \(DEF\). [3]

The scale drawing shows sides \(AB,BC\) of a field. Scale: 5 cm represents 200 m. Use the 1 cm reference below if the image is displayed at a different size.

(a(i)) Measure \(\angle ABC\). [1]

(a(ii)) \(X\) lies on \(BC\) and \(BX=332\) m. Mark \(X\) on the diagram. [2]

(a(iii)) Write the scale as \(1:n\). [2]

(b) A bronze statue is 4.5 m high, mass 195 200 kg, density \(8000\text{ kg/m}^3\). A similar model has volume \(0.385\text{ m}^3\). Find its height. Density = mass ÷ volume. [5]

(a) Find \(h\). [2]

(b) The smaller shape has area \(16\text{ cm}^2\). Find the larger area. [2]

(a) Similar rectangular frames have areas \(350,1134\) square centimetres. The smaller width is 17.5 cm. Find the larger width. [3]

(b) A picture is 15 cm by 10.5 cm, both to the nearest 5 mm. Find the upper bound of its area. [2]

(c) A large frame’s price is reduced 18%. Five frames cost \$166.05 in the sale. Find one frame’s original price. [2]

(d) One frame is advertised for \$57 or 48.20 euros. With \$1 = 0.88 euros, find the difference in dollars to the nearest cent. [3]

(a) \(ABC\) is similar to \(PQR\). \(AB=8,PQ=6,BC=6\) cm. Find \(QR\). [2]

(b) These are cross-sections of similar prisms. The larger volume is \(320\text{ cm}^3\). Find the smaller volume. [2]

(a) In triangle PQR, M lies on QR and S on PR, with the right angles shown. Explain why PMR is similar to MSR, giving reasons. [3]

(b(i)) ABC and PQR are similar, with AC = 4.5 cm, BC = 8.1 cm and QR = 9.9 cm. Find PR = \(x\) cm. [2]

(b(ii)) PQR has area \(25\text{ cm}^2\). Find the area of ABC. [2]

Similar bottles have capacities 324 ml and 768 ml. Smaller height 12 cm. Find larger height. [3]

PQR is similar to ABC, \(PR/AC=2/3\), AB = 9 cm and area ABC = \(18\text{ cm}^2\).

(a) Find PQ. [1]

(b) Find the area of PQR. [2]

Similar objects have volumes \(56\text{ cm}^3\) and \(875\text{ cm}^3\). Smaller height 18 cm. Find the larger height. [3]

A map scale is 1:200000. A lake occupies \(12.4\text{ cm}^2\) on the map. Find its real area in square kilometres. [2]

(a) ABC and PQR are similar. CB = 2.8 cm, RQ = 3.2 cm, CA = 1.61 cm and RP = x cm. Find x. [2]

(b) Similar bowls have capacities 7.8 litres and 4 litres. The larger is 11.5 cm high. Find the smaller height. [3]

A map has scale 1 : 24 000. A lake has area 32 cm² on the map. Find its actual area in km². [2]

For each pair, state whether the triangles are congruent and give the criterion, or write None. [3]

AB ∥ CD; AD and BC intersect at X. AB = 6 cm, CD = 12 cm, CX = 8 cm, DX = 7 cm.

(a) Complete: triangle ABX is … to triangle DCX. [1]

(b) Find BX. [2]

(c(i)) Area of DCX is 26.906 cm². Find area of ABX. [2]

(c(ii)) Find area of ACX. [1]

(a) Find an exterior angle of a regular 18-sided polygon. [2]

(b) B lies on AC, E lies on AD, BE ∥ CD. AE = 5.2 cm, ED = 2.6 cm, CD = 6.75 cm. Find BE. [2]

(c) Similar solids have volumes 32 cm³ and 780 cm³. The smaller height is 2 cm. Find the larger height. [3]

(d) PQ ∥ RS, PNS is straight, and N is the midpoint of RQ. Explain why PQN and SRN are congruent, giving reasons. [4]

The heights are in the ratio small : medium : large = 1 : 5 : 8.

Find shaded area : total unshaded area in simplest form. [4]

A lake has actual area 3 km² and map area 18.75 cm². Find the map scale in the form 1 : n. [3]

PQR is similar to XYZ.

(a(i)) Find YZ. [2]

(a(ii)) XYZ has area 63.6 cm². Find the area of PQR. [3]

(b) Similar containers: the larger has capacity 64.8 litres and surface area 0.792 m²; the smaller has capacity 37.5 litres. Find the smaller surface area. [3]

(a) Manjeet uses 220 litres daily, then reduces this by 15%. Find her new daily usage. [2]

(b) Two bottles are mathematically similar. The large bottle holds 1.35 litres and is 29.7 cm high; the small holds 0.4 litres. Find the smaller height. [3]

(c) A shower supplies 12 litres per minute to a cuboid tank of base 90 cm by 75 cm. Find the water-level increase after 7 minutes. [3]

C lies on a circle with diameter AD. B lies on AC and E on AD, with BE parallel to CD. AB = 21 cm, CD = 18 cm, BE = 13.5 cm.

Find the radius of the circle. [5]

The similar shapes have areas 52.5 cm² and 134.4 cm². Corresponding lengths are m cm and 8 cm. Find m. [3]

A statue model has height4 cm and volume12 cm³. The similar full statue has volume40500 cm³. Find its height. [3]

A paperweight has height4 cm and volume38.4 cm³. A similar paperweight has height7 cm. Find its volume. [3]

A model car has scale 1:20. The actual car has volume 12 m³. Find the volume of the model in cm³. [3]

A map has scale 1:25000. A lake has area 33.6 cm² on the map. Calculate its actual area in km². [2]

FGHJK is an enlargement of regular pentagon ABCDE with centre O.

(a) Find angle AEK. [4]

(b) Areas FGHJK=73.5 cm² and ABCDE=6 cm². Find perimeter FGHJK:perimeter ABCDE in simplest form. [2]

Map scale 1:10000000; Slovakia covers 4.9 cm² on the map. Find its actual area in km². [3]

Two similar containers have heights 30 cm and 75 cm. The larger has capacity 5.5 litres. Calculate the smaller capacity in millilitres. [3]

A, B, C, D lie on the circle. Chords AC and BD intersect at X.

(a) Triangle ADX is ______ to triangle BCX. [1]

(b) Their areas are 36 and 65.61 cm² respectively. AX=8.6 cm and DX=7.2 cm. Find BX. [3]

Triangles T and U are mathematically similar. Corresponding sides are 3 and 12 cm. T has area 5 cm².

Find the area of U. [2]

PQ∥BC; APB and AQC are straight lines. PQ=8 cm, BC=10 cm, AB=9 cm.

(a) Calculate PB. [2]

The similar glasses have capacities 0.5 litres (larger) and 0.25 litres (smaller). Larger height=13 cm.

(b) Calculate the smaller height. [3]

AB=AC and BN=NC.

Complete using a mathematical term: Triangle ABN is ____ to triangle ACN. [1]

AB∥CD; AD⊥BC at X. AB=10, CD=5, AX=8, BX=6 cm.

(a(i)) Use similar triangles to find DX. [2]

(a(ii)) Find XAB. [2]

P,Q,R,S,T lie on the circle centred O. PST=75°, QTS=85°; T,O,Q are collinear.

(b) Find v,w,x,y. [6]

(c) Similar containers have areas 226 and 94 cm². Larger volume 680 cm³. Find smaller volume. [3]

The two barrels are mathematically similar.

The smaller barrel has height h cm and capacity 100 litres. The larger has height 90 cm and capacity 160 litres. Find h. [3]

AB∥CD. AD and BC meet at X. AB=8 cm, CD=4 cm, CX=2 cm, DX=2.8 cm.

(a) Triangle ABX is _____ to triangle DCX. [1]

(b) Calculate AX. [2]

(c) The area of ABX is y cm². Find the area of DCX in terms of y. [1]

Two bottles and their labels are mathematically similar. The smaller bottle contains 0.512 litres and has label area 96 cm². The larger bottle contains 1 litre. Calculate the area of its label. [3]

A model house uses scale 1:30 and has volume 2400 cm³. Find the actual volume in m³. [3]

(a) A cylinder has height 20 cm and cross-sectional area 74 cm². Find its volume. [1]

(b) Similar cylinders A and B have surface areas 440 and 3960 cm². A’s height is 10 cm. Find B’s height. [3]

A map has scale \(1:20000\). A lake has area \(1.6\) cm² on the map.

Find the actual lake area in square metres. [3]

Triangles ABC and PQR are similar, with corresponding vertices in this order. \(AB=5.2\) cm, \(AC=12.4\) cm and \(PR=21.7\) cm.

Find PQ. [2]

A map has scale \(1:1000000\). A forest has area 4.6 cm² on the map.

Calculate the actual area in square kilometres. [2]

Triangles CBA and CED are similar. AB is parallel to DE. \(AB=9\) cm, \(BE=4.8\) cm, \(EC=6\) cm and \(ED=k\) cm.

(a) Find \(k\). [2]

The vases are mathematically similar. Vase A has height 20 cm and volume 1500 cm³. Vase B has volume 2592 cm³.

(b) Calculate \(h\), the height of vase B. [3]

Two cups are mathematically similar. The larger has capacity 0.5 litres and height 8 cm. The smaller has capacity 0.25 litres.

Find the height of the smaller cup. [3]

A backpack of capacity 30 litres has length 53 cm.

Calculate the length of a mathematically similar backpack of capacity 20 litres. [3]

ABC is similar to DEF. AB = 8 cm, BC = 10 cm, AC = y cm. DE = 6 cm, EF = x cm, DF = 9 cm.

(a) Find x. [2]

(b) Find y. [2]

AE and BD are straight lines meeting at C. Angle ABC = angle EDC. ABC and EDC are congruent.

Write two properties of AB and DE. [2]

(a) The two jugs are similar. Find x. [2]

(b) The larger glass has height 16 cm and volume 375 cm³. The smaller has height y cm and volume 192 cm³. Find y. [3]

A map has scale 1 : 50000. A field’s map area is 1.2 cm². Find its actual area in km². [2]

Similar containers have volumes 54 and 128 cm³. The smaller has height 4.5 cm. Find the larger height. [3]

Similar containers have volumes 3456 cm³ and 1458 cm³. The larger surface area is 1024 cm². Find the smaller surface area. [4]

A field plan has scale 1 cm : 15 m. The actual area is 10800 m². Find the plan area. [2]

The cans are similar. The larger holds 1 litre and has diameter 12 cm; the smaller holds 440 ml. Find its diameter d. [3]

A car is 4.4 m long and has a 65-litre fuel tank. A mathematically similar model has a 0.05-litre tank. Find the model’s length in centimetres. [3]

Two similar cones have volumes \(36\pi\) cm³ and \(288\pi\) cm³. The smaller base radius is 3 cm. Find the larger base radius. [3]

\(OP_1=3\) cm, \(P_1P_2=1\) cm, \(\angle OP_1P_2=90^\circ\).

(a) Show \(OP_2=\sqrt{10}\) cm. [1]

\(OP_2P_3\) is mathematically similar to \(OP_1P_2\).

(b(i)) Give \(P_2P_3\) as \(\sqrt a/b\) for integers a and b. [1]

(b(ii)) Give \(OP_3\) as \(c/d\) for integers c and d. [2]

(c) Sidney continues the similar triangles. Find \(OP_5\). [2]

(d(i)) Show \(\angle P_1OP_2\) rounds to 18.4°. [2]

(d(ii)) State \(\angle P_2OP_3\). [1]

(d(iii)) The last triangle before covering the first is \(OP_{n-1}P_n\). Find n. [3]

A model plane has volume 1200 cm³; the full-size plane has volume 4050 m³. Give the model’s linear scale as 1 : n. [3]

A map has scale 1 : 500 000.

(a) Two towns are 172 km apart. Find their map distance in centimetres. [2]

(b) A lake occupies 12 cm² on the map. Find its real area in km². [2]

A model car has linear scale 1/10. Complete the missing entries.

A ship model has scale 1:200 and surface area 7500 cm². Find the ship’s surface area in square metres. [3]

A cereal box measures 10 cm by 25 cm by 35 cm. A mathematically similar special edition has volume 15 120 cm³. Find its dimensions. [3]

A map has scale 1:20 000.

(a) Two points are 2.7 cm apart on the map. Find their actual distance in kilometres. [2]

(b) A field has actual area 64 400 m². Find its map area in cm². [2]

Similar containers have heights 20 cm and 10 cm. The larger has base diameter 9 cm and capacity 1600 ml.

(a) Find the smaller base diameter d. [1]

(b) Find the smaller capacity. [2]

A map has scale 1:250 000.

(a) Find the map distance in centimetres for an actual distance of 80 km. [2]

(b) A forest occupies 6 cm² on the map. Find its actual area in km². [2]

A, B, C and D lie on a circle. AC and BD meet at X.

(a) Explain why BAX = CDX. [1]

AB = 4.40 cm, CD = 9.40 cm, BX = 3.84 cm.

(b)(i) Find CX. [2]

(b)(ii) Triangle ABX has area 5.41 cm². Find the area of CDX. [2]

(a) List all triangles congruent to the shaded triangle. [2]

(b) List all triangles similar to it but not congruent. [2]

A car model has linear scale 1 : 40 and volume 45 cm³. Find the actual car volume in m³. [3]

APB and AQC are straight; PQ ∥ BC. AP = 8 cm, PQ = 10 cm, BC = 12 cm.

Find AB. [2]