# IGCSE - Similarity

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1
(a) Explain why these rectangles are mathematically similar.
..........................................................................................................................................................
..........................................................................................................................................................
[1]
(b) How many times bigger is the area of the large rectangle than the area of the small rectangle?
...................................................
[2]
[Total: 3]
2
B
E
8 cm
NOT TO
SCALE
10 cm
x cm
6 cm
A
y cm
C
D
9 cm
F
Triangle ABC is similar to triangle DEF.
Calculate the value of
(a) x,
[2]
2
(b) y.
[2]
[Total: 4]
3
NOT TO
SCALE
The two containers are mathematically similar in shape.
The larger container has a volume of 3456 cm3 and a surface area of 1024 cm2.
The smaller container has a volume of 1458 cm3.
Calculate the surface area of the smaller container.
[4]
[Total: 4]
4
Paint is sold in cylindrical tins of height 11 cm.
Each tin holds 750 ml of paint.
(a) Write 750 ml in cm3.
[1]
3
(b) Calculate the radius of the tin.
[3]
[2]
(c) A mathematically similar tin has a height of 22 cm.
How many litres of paint does this tin hold?
[Total: 6]
5
Two containers are mathematically similar.
Their volumes are 54 cm3 and 128 cm3.
The height of the smaller container is 4.5 cm.
Calculate the height of the larger container.
[3]
[Total: 3]
4
6
A map is drawn to a scale of 1 : 1 000 000.
A forest on the map has an area of 4.6 cm2.
Calculate the actual area of the forest in square kilometres.
................................................... km2
[2]
[Total: 2]
7
A, B, C and D are points on the circumference of the circle.
AC and BD intersect at X.
(a) Complete the statement.
Triangle ADX is ................................................... to triangle BCX.
[1]
(b) The area of triangle ADX is 36 cm2 and the area of triangle BCX is 65.61 cm2.
AX = 8.6 cm and DX = 7.2 cm.
Find BX.
BX = ................................................... cm
[3]
[Total: 4]
8
A solid hemisphere has volume 230 cm3.
5
(a) Calculate the radius of the hemisphere.
[The volume, V, of a sphere with radius r is
.]
................................................... cm
[3]
(b) A solid cylinder with radius 1.6 cm is attached to the hemisphere to make a toy.
The total volume of the toy is 300 cm3.
(i)
Calculate the height of the cylinder.
................................................... cm
[3]
6
(ii) A mathematically similar toy has volume 19 200 cm3.
Calculate the radius of the cylinder for this toy.
................................................... cm
[3]
[Total: 9]
9
The diagram shows two mathematically similar triangles, T and U.
Two corresponding side lengths are 3 cm and 12 cm.
The area of triangle T is 5 cm2.
Find the area of triangle U.
................................................... cm2
[2]
7
[Total: 2]
10
In the diagram, PQ is parallel to BC.
APB and AQC are straight lines.
PQ = 8 cm, BC = 10 cm and AB = 9 cm.
Calculate PB.
PB = ................................................... cm
[2]
[Total: 2]
8
11
The diagram shows two glasses which are mathematically similar.
The larger glass has a capacity of 0.5 litres and the smaller glass has a capacity of 0.25 litres.
The height of the larger glass is 13 cm.
Calculate the height of the smaller glass.
................................................... cm
[3]
[Total: 3]
12
Two containers are mathematically similar.
The surface area of the larger container is 226 cm2 and the surface area of the smaller container is 94 cm2.
The volume of the larger container is 680 cm3.
Find the volume of the smaller container.
................................................... cm3
[3]
[Total: 3]
9
13
Marianne sells two sizes of photo.
These photos are mathematically similar rectangles.
The smaller photo has length 15 cm and width 12 cm.
The larger photo has area 352.8 cm2.
Calculate the length of the larger photo.
................................................... cm
[3]
[Total: 3]
14
The diagram shows a company logo made from a rectangle and a major sector of a circle.
The circle has centre O and radius OA.
OA = OD = 0.5 cm and AB = 1.5 cm.
E is a point on OC such that OE = 0.25 cm and angle OED = 90&deg;.
10
(a) Calculate the perimeter of the logo.
................................................... cm
[5]
................................................... cm2
[3]
................................................... cm
[3]
(b) Calculate the area of the logo.
(c) A mathematically similar logo is drawn.
The area of this logo is 77.44 cm2.
(i)
Calculate the radius of the major sector in this logo.
11
(ii) A gold model is made.
This model is a prism with a cross-section of area 77.44 cm2.
This gold model is 15 mm thick.
One cubic centimetre of gold has a mass of 19 grams.
Calculate the mass of the gold model in kilograms.
................................................... kg
[3]
[Total: 14]
12
15
The diagram shows two mathematically similar shapes with areas 295 cm2 and 159.5 cm2.
The width of the larger shape is 17 cm.
Calculate the width of the smaller shape.
................................................... cm
[3]
[Total: 3]
13
16
The diagram shows two regular pentagons.
Pentagon FGHJK is an enlargement of pentagon ABCDE, centre O.
The area of pentagon FGHJK is 73.5 cm2.
The area of pentagon ABCDE is 6 cm2.
Find the ratio perimeter of pentagon FGHJK : perimeter of pentagon ABCDE in its simplest form.
.............................. : .............................. [2]
[Total: 2]
14
17
A model of a car has a scale 1 : 20.
The volume of the actual car is 12 m3.
Find the volume of the model.
................................................... cm3
[3]
[Total: 3]
15
18
C lies on a circle with diameter AD.
B lies on AC and E lies on AD such that BE is parallel to CD.
AB = 21 cm, CD = 18 cm and BE = 13.5 cm.
Work out the radius of the circle.
................................................... cm
[5]
[Total: 5]
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