Sec 3 G2 Maths Congruence and similarity worksheet with answers

A free Sec 3 G2 Maths worksheet on congruence and similarity: 8 questions, 30 marks, short questions first and then structured ones, in the school exam style. The answers and full worked solutions are on the last page.

In the print window, choose Save as PDF to download it.

Need the formulas first? Read the Congruence and similarity notes. Or practise it online with step-by-step help when you get one wrong.

Try a question: Congruence and similarity

Free · no sign-up

Answer a real practice question. Get it wrong and MugCorner shows you where it went wrong, with hints and a step-by-step guide.

MugCornerT-Leng Tuition · mugcornersg.com

Sec 3 G2 Maths worksheet: Congruence and similarity

Total: 30 marksSuggested time: 50 minutes

Name: ______________________________Class: ____________ Date: ____________

Answer all questions. Show all your working. Give non-exact answers correct to 3 significant figures, or 1 decimal place for angles in degrees, unless the question says otherwise.

Section A

Triangles

ABC

and

PQR

are congruent, with the equal parts marked. Complete the statement

\triangle ABC \equiv \triangle \ldots

Answer[2]

Two towns are

20

km apart. On a map they are

4

cm apart. Express the scale of the map in the form

1 : n

Answer[2]

In the diagram, the lines

AD

and

BC

meet at

X

, and

AB

is parallel to

DC

. Find

DC

Answercm[3]

(a)A map is drawn to a scale of

1 : 80\,000

. Two places are

12

km apart. Find the distance, in centimetres, between the two places on the map.

Answer (a)cm[2]

(b)On a map,

1

cm represents

0.5

km. Express the scale of the map in the form

1 : n

Answer (b)[2]

Section B

A map is drawn to a scale of

1 : 250\,000

(a)Two towns are

8.8

cm apart on the map. Find the actual distance between them, in kilometres.

Answer (a)km[1]

(b)A road is

4.5

km long. Find its length on the map, in centimetres.

Answer (b)cm[1]

(c)On a second map, the two towns in part (a) are

5

cm apart. Find the scale of the second map in the form

1 : n

Answer (c)[2]

A map is drawn to a scale of

1 : 20\,000

(a)Two towns are

8.8

cm apart on the map. Find the actual distance between them, in kilometres.

Answer (a)km[1]

(b)A road is

4.5

km long. Find its length on the map, in centimetres.

Answer (b)cm[1]

(c)On a second map, the two towns in part (a) are

16

cm apart. Find the scale of the second map in the form

1 : n

Answer (c)[2]

In the diagram,

D

lies on

AB

and

E

lies on

AC

, with

DE

parallel to

BC

AD = 6

cm,

DB = 2

cm,

BC = 9

cm and

AC = 10

cm.

(a)Complete the statement: triangle

ABC

is similar to triangle

\ldots

Answer (a)[1]

(b)Find

DE

Answer (b)cm[2]

(c)Find

EC

Answer (c)cm[2]

In triangle

ABC

D

lies on

AB

and

E

lies on

AC

such that angle

ADE

= angle

ACB

AD = 3

cm,

AE = 5

cm,

EC = 4

cm and

BC = 11

cm.

(a)Complete the statement: triangle

ADE

is similar to triangle

\ldots

Answer (a)[1]

(b)Find the length of

AB

Answer (b)cm[2]

(c)Find the length of

DB

Answer (c)cm[1]

(d)Find the length of

DE

, correct to 3 significant figures.

Answer (d)cm[2]

END OF PAPER

Answers and worked solutions

Question 1

Answer: RPQ[2]

\angle A

(

57^\circ

) matches

\angle \text{R}

\angle B

(

35^\circ

) matches

\angle \text{P}

BC

(

6

cm) matches

\text{PQ}

A

matches

R

B

matches

P

C

matches

Q

\triangle ABC \equiv \triangle \text{RPQ}

Question 2

Answer:

1 : 500\,000

[2]

\mathstrut 4\text{ cm} : 20\text{ km}

4\text{ cm} : 2\,000\,000\text{ cm}

That is

4 : 2\,000\,000

\mathstrut 1 : \dfrac{2\,000\,000}{4}

1 : 500\,000

Question 3

Answer:

DC = 11.25

cm[3]

\angle XAB = \angle XDC

(alternate angles)

\angle AXB = \angle DXC

(vertically opposite angles)

\dfrac{DC}{AB} = \dfrac{XD}{XA}

\mathstrut \dfrac{DC}{5} = \dfrac{9}{4}

DC = \dfrac{9}{4} \times 5

DC = 11.25

Question 4

(a)Answer:

15\text{ cm} : 12\text{ km}

[2]

\mathstrut 1 : 80\,000

\mathstrut 1\text{ cm} : 80\,000\text{ cm}

1\text{ cm} : 0.8\text{ km}

\mathstrut \dfrac{12}{0.8}\text{ cm} : 12\text{ km}

15\text{ cm} : 12\text{ km}

(b)Answer: That is

1 : 50\,000

.[2]

\mathstrut 1\text{ cm} : 0.5\text{ km}

1\text{ cm} : 50\,000\text{ cm}

That is

1 : 50\,000

Question 5

(a)Answer:

22

km[1]

1 : 250\,000

1 \text{ cm} : 250\,000 \text{ cm}

1 \text{ cm} : 2.5 \text{ km}

8.8 \text{ cm} : 22 \text{ km}

(b)Answer:

1.8

cm[1]

1 \text{ cm} : 2.5 \text{ km}

\dfrac{4.5}{2.5} \text{ cm} : 4.5 \text{ km}

1.8 \text{ cm} : 4.5 \text{ km}

(c)Answer:

1 : 440\,000

[2]

5 \text{ cm} : 22 \text{ km}

5 \text{ cm} : 2\,200\,000 \text{ cm}

5 : 2\,200\,000

1 : 440\,000

Question 6

(a)Answer:

1.76

km[1]

1 : 20\,000

1 \text{ cm} : 20\,000 \text{ cm}

1 \text{ cm} : 0.2 \text{ km}

8.8 \text{ cm} : 1.76 \text{ km}

(b)Answer:

22.5

cm[1]

1 \text{ cm} : 0.2 \text{ km}

\dfrac{4.5}{0.2} \text{ cm} : 4.5 \text{ km}

22.5 \text{ cm} : 4.5 \text{ km}

(c)Answer:

1 : 11\,000

[2]

16 \text{ cm} : 1.76 \text{ km}

16 \text{ cm} : 176\,000 \text{ cm}

16 : 176\,000

1 : 11\,000

Question 7

(a)Answer:

ADE

[1]

\angle ADE = \angle ABC

(corresponding angles)

\angle A

is common

\triangle ABC

is similar to

\triangle ADE

(AA)

(b)Answer:

6.75

cm[2]

AB = 6 + 2

\phantom{AB} = 8

\dfrac{DE}{BC} = \dfrac{AD}{AB}

\dfrac{DE}{9} = \dfrac{6}{8}

DE = \dfrac{6}{8} \times 9

\phantom{DE} = 6.75

(c)Answer:

2.5

cm[2]

\dfrac{AE}{AC} = \dfrac{AD}{AB}

AE = \dfrac{6}{8} \times 10

\phantom{AE} = 7.5

EC = 10 - 7.5

\phantom{EC} = 2.5

Question 8

(a)Answer:

ACB

[1]

\angle DAE = \angle CAB

(common ∠)

\angle ADE = \angle ACB

(given)

\triangle ADE

is similar to

\triangle ACB

(AA)

(b)Answer:

15

cm[2]

AC = 5 + 4

\phantom{AC} = 9

\dfrac{AB}{AE} = \dfrac{AC}{AD}

\dfrac{AB}{5} = \dfrac{9}{3}

AB = \dfrac{9 \times 5}{3}

\phantom{AB} = 15

(c)Answer:

12

cm[1]

DB = 15 - 3

\phantom{DB} = 12

(d)Answer:

3.67

cm[2]

\dfrac{DE}{CB} = \dfrac{AD}{AC}

\dfrac{DE}{11} = \dfrac{3}{9}

DE = \dfrac{3 \times 11}{9}

\phantom{DE} = 3.6666\ldots

DE = 3.67

Sec 3 G2 Maths Congruence and similarity worksheet with answers

In the print window, choose Save as PDF to download it.

Need the formulas first? Read the Congruence and similarity notes. Or practise it online with step-by-step help when you get one wrong.

Section B

A map is drawn to a scale of

1 : 250\,000

(a)Two towns are

8.8

cm apart on the map. Find the actual distance between them, in kilometres.

Answer (a)km[1]

(b)A road is

4.5

km long. Find its length on the map, in centimetres.

Answer (b)cm[1]

(c)On a second map, the two towns in part (a) are

5

cm apart. Find the scale of the second map in the form

1 : n

Answer (c)[2]

Answers and worked solutions

Question 1

Answer: RPQ[2]

\angle A

(

57^\circ

) matches

\angle \text{R}

\angle B

(

35^\circ

) matches

\angle \text{P}

BC

(

6

cm) matches

\text{PQ}

A

matches

R

B

matches

P

C

matches

Q

\triangle ABC \equiv \triangle \text{RPQ}

Question 2

Answer:

1 : 500\,000

[2]

\mathstrut 4\text{ cm} : 20\text{ km}

4\text{ cm} : 2\,000\,000\text{ cm}

That is

4 : 2\,000\,000

\mathstrut 1 : \dfrac{2\,000\,000}{4}

1 : 500\,000

Question 3

Answer:

DC = 11.25

cm[3]

\angle XAB = \angle XDC

(alternate angles)

\angle AXB = \angle DXC

(vertically opposite angles)

\dfrac{DC}{AB} = \dfrac{XD}{XA}

\mathstrut \dfrac{DC}{5} = \dfrac{9}{4}

DC = \dfrac{9}{4} \times 5

DC = 11.25

Question 4

(a)Answer:

15\text{ cm} : 12\text{ km}

[2]

\mathstrut 1 : 80\,000

\mathstrut 1\text{ cm} : 80\,000\text{ cm}

1\text{ cm} : 0.8\text{ km}

\mathstrut \dfrac{12}{0.8}\text{ cm} : 12\text{ km}

15\text{ cm} : 12\text{ km}

(b)Answer: That is

1 : 50\,000

.[2]

\mathstrut 1\text{ cm} : 0.5\text{ km}

1\text{ cm} : 50\,000\text{ cm}

That is

1 : 50\,000

Question 5

(a)Answer:

22

km[1]

1 : 250\,000

1 \text{ cm} : 250\,000 \text{ cm}

1 \text{ cm} : 2.5 \text{ km}

8.8 \text{ cm} : 22 \text{ km}

(b)Answer:

1.8

cm[1]

1 \text{ cm} : 2.5 \text{ km}

\dfrac{4.5}{2.5} \text{ cm} : 4.5 \text{ km}

1.8 \text{ cm} : 4.5 \text{ km}

(c)Answer:

1 : 440\,000

[2]

5 \text{ cm} : 22 \text{ km}

5 \text{ cm} : 2\,200\,000 \text{ cm}

5 : 2\,200\,000

1 : 440\,000

Question 6

(a)Answer:

1.76

km[1]

1 : 20\,000

1 \text{ cm} : 20\,000 \text{ cm}

1 \text{ cm} : 0.2 \text{ km}

8.8 \text{ cm} : 1.76 \text{ km}

(b)Answer:

22.5

cm[1]

1 \text{ cm} : 0.2 \text{ km}

\dfrac{4.5}{0.2} \text{ cm} : 4.5 \text{ km}

22.5 \text{ cm} : 4.5 \text{ km}

(c)Answer:

1 : 11\,000

[2]

16 \text{ cm} : 1.76 \text{ km}

16 \text{ cm} : 176\,000 \text{ cm}

16 : 176\,000

1 : 11\,000

Question 7

(a)Answer:

ADE

[1]

\angle ADE = \angle ABC

(corresponding angles)

\angle A

is common

\triangle ABC

is similar to

\triangle ADE

(AA)

(b)Answer:

6.75

cm[2]

AB = 6 + 2

\phantom{AB} = 8

\dfrac{DE}{BC} = \dfrac{AD}{AB}

\dfrac{DE}{9} = \dfrac{6}{8}

DE = \dfrac{6}{8} \times 9

\phantom{DE} = 6.75

(c)Answer:

2.5

cm[2]

\dfrac{AE}{AC} = \dfrac{AD}{AB}

AE = \dfrac{6}{8} \times 10

\phantom{AE} = 7.5

EC = 10 - 7.5

\phantom{EC} = 2.5

Question 8

(a)Answer:

ACB

[1]

\angle DAE = \angle CAB

(common ∠)

\angle ADE = \angle ACB

(given)

\triangle ADE

is similar to

\triangle ACB

(AA)

(b)Answer:

15

cm[2]

AC = 5 + 4

\phantom{AC} = 9

\dfrac{AB}{AE} = \dfrac{AC}{AD}

\dfrac{AB}{5} = \dfrac{9}{3}

AB = \dfrac{9 \times 5}{3}

\phantom{AB} = 15

(c)Answer:

12

cm[1]

DB = 15 - 3

\phantom{DB} = 12

(d)Answer:

3.67

cm[2]

\dfrac{DE}{CB} = \dfrac{AD}{AC}

\dfrac{DE}{11} = \dfrac{3}{9}

DE = \dfrac{3 \times 11}{9}

\phantom{DE} = 3.6666\ldots

DE = 3.67