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Congruence & similarity
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Level 1
Level 2
Level 3
Level 4
Level 5
Congruent triangles & proof
16 questions
Lesson
Not started
Mark as done
In the diagram, the straight lines
P
R
PR
P
R
and
S
Q
SQ
S
Q
cross at
M
M
M
.
P
S
=
Q
R
PS = QR
P
S
=
QR
and angle
S
P
M
SPM
S
P
M
= angle
R
Q
M
RQM
R
QM
.
P
S
Q
R
M
Diagram NOT accurately drawn
Prove that triangle
P
S
M
PSM
P
S
M
is congruent to triangle
Q
R
M
QRM
QR
M
.
●●●●
●
Level 4
3 marks
Start
→
Mark as done
A
B
C
D
ABCD
A
B
C
D
is a rectangle. The diagonal
B
D
BD
B
D
is drawn.
A
B
C
D
Prove that triangle
A
B
D
ABD
A
B
D
is congruent to triangle
C
D
B
CDB
C
D
B
.
●●●●
●
Level 4
3 marks
Start
→
Mark as done
K
L
M
N
KLMN
K
L
M
N
is a kite, with
K
L
=
K
N
KL = KN
K
L
=
K
N
and
M
L
=
M
N
ML = MN
M
L
=
M
N
. The diagonal
K
M
KM
K
M
is drawn.
K
L
M
N
(a)
Prove that triangle
K
L
M
KLM
K
L
M
is congruent to triangle
K
N
M
KNM
K
N
M
.
(b)
Hence explain why the diagonal
K
M
KM
K
M
bisects angle
L
K
N
LKN
L
K
N
.
●●●●
●
Level 4
4 marks
Start
→
Similar shapes & missing lengths
28 questions
Lesson
Not started
Mark as done
In the diagram,
D
E
DE
D
E
is parallel to
A
B
AB
A
B
.
A
D
=
9
AD = 9
A
D
=
9
cm,
D
C
=
6
DC = 6
D
C
=
6
cm,
D
E
=
4
DE = 4
D
E
=
4
cm and
A
B
=
10
AB = 10
A
B
=
10
cm.
10 cm
4 cm
9 cm
6 cm
A
B
C
D
E
Diagram NOT accurately drawn
(a)
Explain why triangle
C
D
E
CDE
C
D
E
is similar to triangle
C
A
B
CAB
C
A
B
.
(b)
Work out the length of
C
B
CB
C
B
, given that
C
E
=
5
CE = 5
C
E
=
5
cm.
●●●●
●
Level 4
5 marks
Start
→
Mark as done
Two bottles are mathematically similar. Their heights are
12
12
12
cm and
18
18
18
cm.
The smaller bottle holds
500
500
500
ml.
Work out how much the larger bottle holds.
●●●●
●
Level 4
3 marks
Start
→
Mark as done
Two mathematically similar rugs have areas
45
45
45
m² and
125
125
125
m².
The shorter side of the smaller rug is
6
6
6
m.
Work out the shorter side of the larger rug.
●●●●
●
Level 4
3 marks
Start
→
Mark as done
Two mathematically similar boxes have heights
10
10
10
cm and
15
15
15
cm.
The smaller box has volume
400
400
400
cm³.
Work out the volume of the larger box.
●●●●
●
Level 4
3 marks
Start
→
Mark as done
In triangle
A
B
C
ABC
A
B
C
, the point
D
D
D
lies on
A
B
AB
A
B
and the point
E
E
E
lies on
A
C
AC
A
C
, with
D
E
DE
D
E
parallel to
B
C
BC
B
C
.
A
D
=
4
AD = 4
A
D
=
4
cm,
D
B
=
6
DB = 6
D
B
=
6
cm and
D
E
=
6
DE = 6
D
E
=
6
cm.
A
B
C
D
E
Diagram NOT accurately drawn
Work out the length of
B
C
BC
B
C
.
●●●●
●
Level 4
3 marks
Start
→
Mark as done
Rectangle
R
R
R
has vertices
(
2
,
2
)
(2, 2)
(
2
,
2
)
,
(
5
,
2
)
(5, 2)
(
5
,
2
)
,
(
5
,
4
)
(5, 4)
(
5
,
4
)
and
(
2
,
4
)
(2, 4)
(
2
,
4
)
.
Rectangle
R
R
R
is enlarged by scale factor
3
3
3
, centre the origin, to give rectangle
R
′
R'
R
′
.
(a)
Work out the perimeter of rectangle
R
′
R'
R
′
.
(b)
Work out how many times bigger the area of
R
′
R'
R
′
is than the area of
R
R
R
.
(c)
Rectangle
R
′
R'
R
′
is then translated by the vector
(
−
4
1
)
\begin{pmatrix} -4 \\ 1 \end{pmatrix}
(
−
4
1
)
. Write down the coordinates of the image of the vertex
(
6
,
6
)
(6, 6)
(
6
,
6
)
of
R
′
R'
R
′
.
●●●●
●
Level 4
5 marks
Start
→
Mark as done
Two solid cylinders,
P
P
P
and
Q
Q
Q
, are mathematically similar.
The height of
P
P
P
is
6
6
6
cm and the height of
Q
Q
Q
is
9
9
9
cm. The volume of
P
P
P
is
40
40
40
cm
3
^3
3
.
Work out the volume of
Q
Q
Q
. (3 marks)
●●●●
●
Level 4
3 marks
Start
→
Mark as done
Two solid cones,
A
A
A
and
B
B
B
, are mathematically similar.
The height of
A
A
A
is
4
4
4
cm and the height of
B
B
B
is
6
6
6
cm.
(a)
The surface area of
A
A
A
is
32
32
32
cm
2
^2
2
. Work out the surface area of
B
B
B
. (2 marks)
(b)
The volume of
B
B
B
is
405
405
405
cm
3
^3
3
. Work out the volume of
A
A
A
. (3 marks)
●●●●
●
Level 4
5 marks
Start
→
Mark as done
Two solids
A
A
A
and
B
B
B
are mathematically similar.
The surface area of solid
A
A
A
is
45
45
45
cm
2
^2
2
. The surface area of solid
B
B
B
is
80
80
80
cm
2
^2
2
.
The volume of solid
A
A
A
is
135
135
135
cm
3
^3
3
.
Work out the volume of solid
B
B
B
. (4 marks)
●●●●
●
Level 4
4 marks
Start
→
Mark as done
In the diagram,
D
E
DE
D
E
is parallel to
A
B
AB
A
B
.
A
D
=
6
AD = 6
A
D
=
6
cm,
D
C
=
4
DC = 4
D
C
=
4
cm and
D
E
=
6
DE = 6
D
E
=
6
cm.
6 cm
4 cm
6 cm
A
B
C
D
E
Diagram NOT accurately drawn
(a)
Work out the length of
A
B
AB
A
B
. [3]
(b)
Write down the ratio of the area of triangle
C
D
E
CDE
C
D
E
to the area of triangle
C
A
B
CAB
C
A
B
in its simplest form. [1]
●●●●
●
Level 4
4 marks
Start
→
Mark as done
Two solid cuboids are mathematically similar. Their surface areas are
36
36
36
cm
2
^2
2
and
100
100
100
cm
2
^2
2
.
The volume of the smaller cuboid is
54
54
54
cm
3
^3
3
.
Work out the volume of the larger cuboid. [4]
●●●●
●
Level 4
4 marks
Start
→
Mark as done
Two vases are mathematically similar.
The smaller vase has a volume of
120
120
120
cm³ and a height of
9
9
9
cm.
The larger vase has a volume of
840
840
840
cm³ and a height of
H
H
H
cm.
(a)
Work out the value of
H
H
H
.
Give your answer correct to 1 decimal place.
(4 marks)
The surface area of the smaller vase is
A
A
A
cm²
The surface area of the larger vase is
p
A
pA
p
A
cm²
(b)
Find the exact value of
p
p
p
.
(1 mark)
●●●●
●
Level 4
5 marks
Start
→
Mark as done
P
Q
R
PQR
P
QR
is a triangle.
P
Q
R
S
T
P
S
Q
PSQ
P
S
Q
and
P
T
R
PTR
P
T
R
are straight lines.
S
T
ST
S
T
is parallel to
Q
R
QR
QR
.
(a)
Prove that triangle
P
Q
R
PQR
P
QR
is similar to triangle
P
S
T
PST
P
S
T
.
(2 marks)
S
T
=
12
ST = 12
S
T
=
12
cm \qquad
Q
R
=
20
QR = 20
QR
=
20
cm \qquad
P
R
=
35
PR = 35
P
R
=
35
cm
(b)
Work out the length of
T
R
TR
T
R
.
(3 marks)
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●
Level 4
5 marks
Start
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