�3 12.1 Congruent triangles

297A

1 Are these two triangles congruent? Justify your answer.

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2 Are these two triangles congruent? Justify your answer.

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3 Are these two triangles congruent? Justify your answer.

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70˚

80˚ 30˚

70˚

8 cm 8 cmA

C

B R Q

P

70˚

80˚ 30˚

70˚

8 cmA

C

B R

8 cm

Q

P

8 cm 8 cm

30˚B R Q

PC

A8 cm 8 cm

30˚B R Q

PC

A

Guided practice worksheet

A

Questions are targeted at the grades indicated

�3 12.1 Congruent triangles

297B

Guided practice worksheet

4 Are these two triangles congruent? Justify your answer.

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5 Are these two triangles congruent? Justify your answer.

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8 cm 8 cm

10 cm 10 cm

C

A B R Q

P

8 cm

10 cm

C

A B30°

8 cm

10 cm

R Q

P

30°

You must not assume that if two triangles look the same, they are congruent

A

�3 12.2 Recognising similar shapes

299

1 Which pairs of shapes are similar?

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2 The rectangles shown in the diagram are similar.

Calculate the value of the side a.

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3 Are rectangles A and B similar?

Explain your answer.

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4 A photograph is 12 cm long and 8 cm wide. It is mounted on a piece of card so that there is a 4 cm border around the photograph.

Are the photograph and the card similar shapes? Explain your answer.

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Guided practice worksheet

Do not forget the corresponding angles must be equal.

45˚

45˚

H

60˚

60˚ 60˚

GF

E

A C DB

A

Questions are targeted at the grades indicated

D

2.3 cm

4.8 cm

6.2 cm

BA12 cm

1.6 cm

7.2 cm

a cm

23.04 cm

Remember to check which sides are corresponding.

B

�3 12.3 Similar triangles

301A

1 Triangles ABC and DEF are similar.

a Write down three pairs of corresponding sides.

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b Calculate the length x.

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c Calculate the length y.

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2 a Prove that triangle AMN is similar to triangle ABC.

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b Calculate the length AB.

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c Calculate the length MB.

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3 a Prove that triangle PQR is similar to triangle STR.

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b Calculate the length of SR.

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Guided practice worksheet

ABCWrite the equation with sides from –––– DEF

Use the facts you know about parallel lines to prove corresponding angles are equal.

A

C

B

M

N

6 cm

4.8 cm

12 cm

80°

60°

10 cm

3 cm

7.5 cm ED

B

A

x y

C F

40°

40°

8 cm

3 cm4 cm

S

P Q

T

R

A

A*

Questions are targeted at the grades indicated

�3 12.3 Similar triangles

301B

Guided practice worksheet

4 a Prove that triangle ABM is similar to triangle CDM.

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b Calculate the length of side MC.

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c Calculate the length of side MD.

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5 a Prove that triangle ABC is similar to triangle CDE.

AE has length 15 cm.

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b Calculate the length of side AC.

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c Calculate the length of side CE.

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Identify the corresponding sides before you begin.

5 cm

7 cm

A

C

D E

B

A*

A

B

M4.2 cm

4.6 cm3 cm

18 cm

C

D

Identify the corresponding sides before you begin.

�3 12.4 Areas of similar shapes

303A

1 Pentagons A and B are similar with scale factor 3.

a Calculate the area scale factor.

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The area of A is 9 cm2.

b Calculate the area of B.

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2 P and Q are regular hexagons.

a Calculate the scale factor.

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b Calculate the area scale factor.

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c The area of P is 13.5 cm2. Calculate the area of Q.

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3 A rubber ball has a radius of 5 cm. The surface area of the ball is 314 cm2.

A similar ball has a radius of 8 cm. Calculate the surface area of the larger ball.

(Give your answer to the nearest cm2.)

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4 Cylinders P and Q are similar. The surface area of cylinder P is 50 cm2.

The surface area of cylinder Q is 1800 cm2. The radius of P is 4 cm.

Calculate the radius of Q.

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Guided practice worksheet

length × length gives area. The scale factor must be squared.

AB

P

3 cm 4.5 cm

Q

A

Questions are targeted at the grades indicated

�3 12.4 Areas of similar shapes

303B

Guided practice worksheet

5 These soup tins are similar.

The area of the label for the large tin is 4 × as big as the label for the small tin.

How tall is the large tin?

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6 The leaves on a tree are similar.

James measured the leaves as part of a science experiment.

He estimated the area of the smaller leaf to be 20 cm2 using a cm squared grid.

Estimate the area of the larger leaf.

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7 Two cubes are dipped in gold.

The smaller cube has length 2 cm and the larger cube has length 5 cm.

How many times as much gold is needed for the larger cube?

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Use the area scale factor to � nd the scale factor.

12 cm7 cm

A

7 cmhSoup

Soup

�3 12.5 Volumes of similar shapes

305A

1 The diagram shows two bottles.

The large bottle contains 250 cm3.

Calculate the volume of the small bottle.

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2 A suitcase’s height is 40 cm. A similar suitcase from the same set has height 60 cm. If the small suitcase has volume 40 000 cm3, � nd the volume of the larger suitcase.

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3 The pentagonal prism has length 5 cm. Its volume is 50 cm3. A similar larger prism has length 7 cm.

Calculate the volume of the larger prism.

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4 A � sh with a mass of 1.4 kg is 0.5 m long.

Calculate the expected mass of a similar fish whose length is 0.6 m.

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5 Two metal spheres have masses of 4 kg and 108 kg respectively.

If the radius of the small sphere is 5 cm find the radius of the larger sphere.

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Guided practice worksheet

length × length × length gives a volume. The scale factor must be cubed.

12 cm

20 cm

Work out the volume scale factor and use it to � nd the mass.

A

Questions are targeted at the grades indicated

�3 12.5 Volumes of similar shapes

305B

Guided practice worksheet

6 A company sell bins for recycling. The bins are mathematically similar.

A 16-litre bin has height 30 cm. Calculate the height of the 60-litre bin.

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7 These funnels are mathematically similar.

The capacity of the larger funnel is 1350 ml and the capacity of the smaller funnel is 50 ml.

The height of the smaller funnel is 6 cm.

Calculate the height of the larger funnel.

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8 A company provides gift boxes in three sizes. The boxes are geometrically similar.

The height of the small box is 10 cm and the height of the medium box is 20 cm.

The volume of the medium box is 2400 cm3.

a Calculate the volume of the small box.

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b The large box has a volume of 64 800 cm3.

Calculate the height of the large box.

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6 cm

A

�3 12.6 Lengths, areas and volumes of similar shapes

307A

1 Two similar cones have their surface areas in the ratio 9:16.

Find the ratio of

a the lengths

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b the volumes.

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2 Two solid spheres have surface areas of 8 cm2 and 72 cm2 respectively.

The mass of the smaller sphere is 2 kg. Find the mass of the larger sphere.

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3 The volume of a prism is 120 cm3. The volume of a larger prism is 405 cm3. If the larger prism has a surface area of 315 cm2, calculate the surface area of the smaller prism.

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4 The areas of the bases of two glasses are in the ratio 4:9.

a Find the ratio of their volumes.

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b If the smaller glass holds 124 cm3 of liquid, how much will the large glass hold?

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5 A small bag that was carried as hand luggage has a length of 50 cm and a weight of 10 kg. A similar bag of weight 18 kg is placed in the hold. Calculate the length of this bag.

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Work out the volume scale factor and use it to � nd the mass.

Guided practice worksheet Questions are targeted at the grades indicated

A

�3 12.6 Lengths, areas and volumes of similar shapes

307B

6 The ingredients of a bar of chocolate cost 25 p. A giant sized bar is similar in shape, but � ve times longer than the original.

Calculate the cost of the ingredients of the giant sized bar.

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7 A small cone is made by cutting the top from a large cone.

The area of the base of the large cone is 100 cm2.

The area of the base of the smaller cone is 4 cm2.

The volume of the large cone is 750 cm3.

Find the volume of the smaller cone.

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8 The lid of a treasure chest has an area of 1800 cm2.

A larger similar box has a lid of area of 11 250 cm2. If the volume of the larger chest is 312 500 cm3, find the volume of the smaller chest.

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9 The surface area of a large sphere is 64 times the surface area of a smaller sphere. The volume of the small sphere is 50 cm3.

Calculate the radius of the large sphere.

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The volume of a sphere is 4–3 πr3.

Guided practice worksheet

A*

�3 12.6 Lengths, areas and volumes of similar shapes

307C

Guided practice worksheet

1 0 Frustums A and B are similar.

The volume of frustum A is 162π cm3.

The volume of frustum B is 384π cm3.

The surface area of A is 288 cm2.

Calculate the surface area of frustum B.

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B A

A*