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21 questions at your level
Difficulty
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Level 2
Level 3
Level 4
Level 5
Ruler-and-compass constructions
10 questions
Lesson
Not started
Mark as done
(a)
Draw a straight line
A
B
AB
A
B
of length
8.4
8.4
8.4
cm.
(b)
Using ruler and compasses only, construct the perpendicular bisector of
A
B
AB
A
B
. You must show all your construction arcs.
(c)
Write down the distance from
A
A
A
to the point where your perpendicular bisector crosses
A
B
AB
A
B
.
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Level 2
4 marks
Start
→
Mark as done
(a)
Using ruler and compasses only, construct an equilateral triangle with sides of length
6
6
6
cm. You must show all your construction arcs.
(b)
Write down the size of each angle of your triangle.
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Level 2
3 marks
Start
→
Mark as done
The diagram shows an angle
P
Q
R
PQR
P
QR
.
R
Q
P
Using ruler and compasses only, construct the bisector of angle
P
Q
R
PQR
P
QR
. You must show all your construction arcs.
●●●
●●
Level 3
3 marks
Start
→
Mark as done
The diagram shows a straight line
l
l
l
, a point
P
P
P
which is not on the line, and a point
Q
Q
Q
which is on the line.
P
Q
l
(a)
Using ruler and compasses only, construct the perpendicular from
P
P
P
to the line
l
l
l
. You must show all your construction arcs.
(b)
Using ruler and compasses only, construct the perpendicular to
l
l
l
at the point
Q
Q
Q
.
(c)
Aisha says "the perpendicular I drew in part (a) is the shortest path from
P
P
P
to the line". Explain why Aisha is right.
●●●
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Level 3
6 marks
Start
→
Mark as done
(a)
Draw a straight line and mark a point
P
P
P
on it. Using ruler and compasses only, construct an angle of
90
°
90°
90°
at
P
P
P
.
(b)
By adding one further construction, obtain an angle of
45
°
45°
45°
at
P
P
P
. Show all your construction arcs.
●●●
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Level 3
4 marks
Start
→
Mark as done
(a)
Draw a line segment
M
N
MN
M
N
of length
7.6
7.6
7.6
cm. Using ruler and compasses only, construct the perpendicular bisector of
M
N
MN
M
N
.
(b)
Your bisector crosses
M
N
MN
M
N
at the point
X
X
X
. Write down the length of
M
X
MX
M
X
.
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Level 3
3 marks
Start
→
Mark as done
(a)
Draw a straight line and mark a point
X
X
X
on it.
Using ruler and compasses only, construct an angle of
60
°
60°
60°
at
X
X
X
. You must show all your construction arcs.
(b)
Hence construct an angle of
30
°
30°
30°
.
(c)
Explain why the construction in part (a) gives exactly
60
°
60°
60°
.
●●●●
●
Level 4
5 marks
Start
→
Mark as done
Elin bisects the angle at
Q
Q
Q
using the standard construction: an arc centred at
Q
Q
Q
cuts the two arms at points
S
S
S
and
T
T
T
, then equal arcs centred at
S
S
S
and at
T
T
T
cross at the point
U
U
U
, and she draws the line
Q
U
QU
Q
U
.
Explain
why
this construction bisects the angle. You should refer to the triangles
Q
S
U
QSU
QS
U
and
Q
T
U
QTU
QT
U
.
●●●●
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Level 4
3 marks
Start
→
Mark as done
(a)
Using ruler and compasses, construct triangle
A
B
C
ABC
A
B
C
with
A
B
=
7
AB = 7
A
B
=
7
cm,
B
C
=
6
BC = 6
B
C
=
6
cm and
A
C
=
5
AC = 5
A
C
=
5
cm.
(b)
Construct the perpendicular bisector of
B
C
BC
B
C
and the bisector of angle
B
A
C
BAC
B
A
C
. Extend both constructed lines far enough for them to cross, and mark the crossing point
X
X
X
. (You may find that
X
X
X
lies outside the triangle.)
(c)
Ravi says "
X
X
X
is the same distance from
B
B
B
as from
C
C
C
, and it is also the same distance from
A
B
AB
A
B
as from
A
C
AC
A
C
." Explain why Ravi is right.
●●●●●
Level 5
6 marks
Start
→
Mark as done
Using ruler and compasses only, construct an angle of
75
°
75°
75°
. You must show all your construction arcs.
Hint:
75
=
60
+
15
75 = 60 + 15
75
=
60
+
15
.
●●●●●
Level 5
4 marks
Start
→
Loci & regions
11 questions
Lesson
Not started
Mark as done
A goat is tied to a post
P
P
P
in the middle of a large field by a rope of length
9
9
9
m.
A plan of the field is drawn using a scale of
1
1
1
cm to
2
2
2
m.
(a)
Work out the radius, in centimetres, of the circle that should be drawn on the plan to show how far the goat can reach.
(b)
Describe the region of the plan that represents the grass the goat can eat.
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Level 2
3 marks
Start
→
Mark as done
A plan is drawn to a scale of
1
1
1
cm to
5
5
5
m.
(a)
A water sprinkler reaches every point within
20
20
20
m of the tap
T
T
T
. Work out the radius of the circle that shows this region on the plan.
(b)
Describe fully the locus of points that are exactly
20
20
20
m from
T
T
T
(in the real garden).
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Level 2
4 marks
Start
→
Mark as done
The diagram shows a plan of a rectangular shed measuring
30
30
30
m by
15
15
15
m, drawn to a scale of
1
1
1
cm to
5
5
5
m.
30 m
15 m
shed
A security light lights up all the ground within
20
20
20
m of the outside walls of the shed.
On the plan, draw accurately the locus of the points that are exactly
20
20
20
m from the outside of the shed.
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Level 3
4 marks
Start
→
Mark as done
A treasure map is drawn to a scale of
1
1
1
cm to
4
4
4
m.
On the map, the treasure is buried at the point that is equidistant from a rock
R
R
R
and a well
W
W
W
, and also equidistant from two straight paths that meet at a point
T
T
T
(measured within the acute angle between the paths).
(a)
Name the construction needed for each of the two conditions.
(b)
A student measures the distance from
R
R
R
to the treasure on the map as
5.5
5.5
5.5
cm. Work out the real distance from the rock to the treasure.
●●●
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Level 3
4 marks
Start
→
Mark as done
Two statues,
A
A
A
and
B
B
B
, stand
12
12
12
m apart in a park. A fountain is to be built so that it is exactly the same distance from
A
A
A
as from
B
B
B
.
(a)
Describe fully the locus of possible positions for the fountain.
A plan of the park is drawn to a scale of
1
1
1
cm to
2
2
2
m.
(b)
Work out the length of
A
B
AB
A
B
on the plan.
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Level 3
3 marks
Start
→
Mark as done
A
B
C
D
ABCD
A
B
C
D
is a rectangular room, with the corner
A
A
A
where walls
A
B
AB
A
B
and
A
D
AD
A
D
meet.
A sensor must be placed so that it is
closer to wall
A
B
AB
A
B
than to wall
A
D
AD
A
D
.
(a)
Describe the boundary of the region where the sensor can go, and how you would construct it.
The room measures
8
8
8
m by
6
6
6
m. A plan is drawn to a scale of
1
1
1
cm to
2
2
2
m.
(b)
Work out the dimensions of the room on the plan.
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Level 3
4 marks
Start
→
Mark as done
The point
D
D
D
is shown on a centimetre grid.
A point
Q
Q
Q
is no more than
3
3
3
cm from the point
D
D
D
.
Kofi is asked to shade the region where the point
Q
Q
Q
could be.
His answer is shown on the grid.
10
12
14
16
18
20
22
11
12
13
14
15
16
17
18
19
20
21
D
Explain the mistake Kofi has made.
(1 mark)
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Level 3
1 mark
Start
→
Mark as done
The diagram shows a plan of a field with a straight fence and two trees,
A
A
A
and
B
B
B
. The plan is drawn to a scale of
1
1
1
cm to
10
10
10
m.
A
B
fence
A bird table is to be placed so that it is:
* less than
45
45
45
m from tree
A
A
A
,
and
* nearer to tree
A
A
A
than to tree
B
B
B
,
and
* more than
20
20
20
m from the fence.
On the plan, show by shading the region where the bird table can be placed. You must show all your construction lines.
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●
Level 4
5 marks
Start
→
Mark as done
A guard dog is attached by a
5
5
5
m chain to a ring that slides freely along a straight rail of length
20
20
20
m, fixed to the ground in a large open yard.
20 m
Diagram NOT accurately drawn
(a)
Describe the region of the yard the dog can reach.
(b)
Work out the area of this region. Give your answer correct to 1 decimal place.
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●
Level 4
5 marks
Start
→
Mark as done
P
Q
R
S
PQRS
P
QR
S
is a rectangular garden measuring
16
16
16
m by
10
10
10
m. The plan below is drawn to a scale of
1
1
1
cm to
2
2
2
m.
16 m
10 m
P
Q
R
S
A pond is to be built in the garden so that it is:
* more than
12
12
12
m from corner
P
P
P
,
and
* less than
6
6
6
m from the side
Q
R
QR
QR
,
and
* nearer to the side
P
Q
PQ
P
Q
than to the side
S
R
SR
S
R
.
On the plan, show by shading the region where the pond can be built. You must show all your construction lines.
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Level 5
5 marks
Start
→
Mark as done
A goat is tied by a
10
10
10
m rope to the corner
G
G
G
of a shed which measures
8
8
8
m by
6
6
6
m. The shed stands in a large open field, and the goat cannot go inside the shed.
8 m
6 m
G
shed
Diagram NOT accurately drawn
Work out the total area of the field the goat can graze. Give your answer correct to 3 significant figures.
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Level 5
4 marks
Start
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