Student Booklet
General atg. s
Specialist
Circle Ge • etry
Mensuration
Contents: Maths OnLine Lessons & Work Sheets
1. Definitions and Terms
2. Chord Properties
3. Angles at the Centre
4. Angles on the same Arc
5. Cyclic Quadrilaterals
6. Angles in the Semi Circle
7. Tangent Properties
8. Mensuration Work Sheet
Definitions and Terms - Worksheet
1
Draw a clear sketch to illustrate the following information in each question,
using circles with centre 0 unless otherwise instructed.
Q1
A, B and C are concylclic.
Q2
Shade in the minor segment formed by the chord XY.
Q3
The arc PQ subtends an angle of 100° at the centre.
Q4 The arc IVIN subtends an angle of 40° at a point P on the circumference.
Q5 AB and CD are two chords intersecting at X.
Q6
PQRS is a cyclic quadrilateral.
Q7
ST is a tangent touching the circle at T. SAB is a secant cutting the circle at A and B.
Q8
AB is a common chord to two circles with centres 0 and C respectively.
Q9
PTQ is a common tangent to two circles, centres 0 and C, that touch externally.
Q10
The diameter AB subtends an angle at a point C on the circumference.
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Page 1
Chord Properties - Worksheet
2
Give answers correct to 1 decimal place where necessaly.
Q1
Q5 A chord AB is drawn 7cm from the centre
of a circle with radius 12cm. Calculate the
length of AB.
Q6
ANSWERS
A) 3 cm
B) 4 cm
Q2
C) 4.8 cm
D) 5 cm
E) 6cm
F) 7.2 cm
AB = 8cm. C is the midpoint
of AB. If the circle has radius 5cm,
find the length of OC.
A chord is drawn 24cm from the
centre of concentric circles with
radii 26cm and 30cm.
Find the length of PR.
H) 8 cm
I) 8.4 cm
J) 9.2 cm
K) 9.7 cm
Q7
Q3
G) 7.8cm
L) 10.1cm
M) 10.7cm
N) 12 cm
0) 13 cm
P) 15 cm
The chord XY is 24cm long and
is 5cm from the centre 0.
Calculate the radius of the circle.
Q4
PAB is an isosceles triangle
inscribed in a circle of radius 26cm.
M is the midpoint of AB.
If PA = PB and AB = 48cm,
calculate the length PM.
Q) 16.8 cm
R) 18 cm
S) 19.5cm
T) 21.4cm
U) 22 cm
V) 28 cm
W) 30 cm
X) 32 cm
Y) 36cm
OC = 9cm. Find the length of AB.
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Z) 44 cm
Page 1
Angles at the Centre - Worksheet
3
Find the value of x in each diagram.
Q1
ANSWERS
A) 17°
Q5
B) 30°
C) 34°
D) 38°
E) 41 °
F) 45°
G) 46°
Q2
Q6
H) 50°
I) 52°
J) 58°
K) 70°
L) 72°
M) 82°
N) 108 °
Q3
° 0)1
Q7
P) 112°
Q) 116°
R) 124°
S) 134°
T) 138 °
U) 148 °
V) 152°
Q4
Q8
W) 156 °
X) 164°
B
Y) 170°
Z) 220°
A0//CB and LA = 24 ° .
Find x.
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Page 1
Angles on the Same Arc - Worksheet
4
Find the value of x in each diagram.
Q1
Q5 AABC is equilateral.
ANSWERS
A) 28 °
B) 30 °
C) 32 °
D) 35 °
E) 36 °
F) 38 °
G) 40 °
Q2
Q6
H) 42 °
I) 45 °
J) 46°
K) 48 °
L) 50 °
M) 51 °
N) 53 °
° 0)54
Q3
P) 58°
Q) 60 °
R) 72 °
S) 80 °
T) 84°
U) 90 °
V) 93 °
Q4
Q8
W) 97 °
X) 98 °
Y) 102 °
Z) 105°
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Page 1
Cyclic Quadrilaterals - Worksheet
5
Find the value of x in each diagram.
Q1
(X
ANSWERS
A) 28 °
B) 30 °
C) 31 °
D) 34 °
E) 35 °
F) 38 °
G) 40 °
Q2
Q6
H) 42 °
I) 51 °
J) 54 °
K) 57 °
L) 60 °
M) 62°
N) 65 °
Q3
Q7
° 0)67
P) 72 °
Q) 74°
R) 75 °
S) 85 °
T) 86 °
U) 92 °
V) 94 °
Q4
el0
W) 105°
X) 115 °
Y) 121 °
Z) 136°
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Page 1
Angle in a Semi-Circle - Worksheet
6
Find the value ofx in each diagram, correct to 1 decimal place where necessary.
Q1
Qf
ANSWERS
A) 26°
B) 27°
C) 28°
D) 30°
E) 32°
F) 38°
G) 42°
Q2
Q6
H) 43°
1)44°
J) 45 °
K) 46°
L) 48°
M) 54°
N) 59°
Q3
Q7
°0)85
P) 90°
Q) 92°
R) 115°
S) 4.6cm
T) 5.2cm
U) 6cm
V) 7.4cm
Q4
Q8
W) 8cm
X) 9cm
Y) 9.8cm
Z) 11.1cm
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Page 1
Tangent Properties Worksheet
7
Find the value of the pronumeral in each diagram, correct to 1 decimal place where necessary.
Qi
20cm
Q5
ANSWERS
Q6
68'
0
•
A) 4 cm
N) 16.8cm
B) 4.7 cm
0) 17.7cm
C) 5 cm
P) 17.9 cm
D) 5.1 cm
Q) 18.2cm
E) 7 cm
R) 21.5 cm
F) 8 cm
S) 23.3 cm
G) 8.9 cm
T) 24.2cm
H) 9.3 cm
U) 74 cm
I) 10 cm
V) 80 cm
J) 10.6 cm W) 82 cm
K) 11 cm
X) 44°
L) 11.5 cm Y) 64°
M) 15.5 cm Z) 68°
Q3 PT is a tangent to the circle touching
it at T. PT = 8cm and PO = 12cm.
Find the length of diameter AB.
Q7
10cm
7cm
0
OC = 18cm. Find x.
Q4
12cm
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Page 1
Mensuration Work sheet 8
I. Calculate, in degrees(2 dit.t....4), the magnitude of the angle subtended at the
centre of a circle of radius length 8 cm by an arc whose length is 15 cm.
2. Find the length of an arc of a ci'rcle of radius 15 cm if the arc subtends an angle
of 70° at the centre.
3. An arc of a circle subtends an angle of 52 ° at the circumference. lithe radius
length of the circle is 12 cm, calculate the arc length.
4. The area °fa sector OA B of a circle, centre 0, radius length 20 cm is 240 cmz.
Calculate
(a) the magnitude or the angle 408,
(b) the length of arc AB,
(c) the length of the chord AB.
5. A chord PQ, 24 cm long, is 5 cm from the centre of the circle. Calculate the
length of the arc PQ.
6. A point P is 8 cm distant from the centre of a circle of radius length 5 cm. Find
the length of the major arc between the points of contact of the tangents drawn
from P to the circle.
7. A chord AB of a circle with centre 0 has length 16 cm. lithe radius of the circle
is 10 cm, calculate
(a) the magnitude of angle A08,
(b) the length of the minor arc AB.
(c) the area of the minor segment formed by the chord AB.
8. The minute hand of a clock is 20 cm long. Calculate
(a) the arc length along which the tip of the hand travels in 16 min.
(b) the shortest distance between the initial and final positions of the tip of the
hand.
9. An arc AB subtends an angle of 06.271 vat the centre of a circle of radius 15 Ctn.
Calculate the difference in the lengths of the chord AS and the minor arc AB.
10. From a circular piece of metal 6 cm in diameter, a sector of angle 30' is removed.
Find the area remaining. Express your answer in terms of n.
II. A chord subtends an angle of 126-).06 at the centre of a circle of radius 16 cm.
Find the difference in length between the chord and the arc.
12. Two circles of radii 6 cm and 8 cm have their centres 10 cm apart. Calculate the
length of the arc of the smaller circle cut off by the larger circle.
13. The minute hand of a clock is 3 cm in length. What area is swept out by the
hand in an interval of 40 min? Express your answer in terms of n.
14. A circular metal plate is cut into two segments along a chord equal in length to
the radius. What is the ratio of the areas of the two segments?
15. A sheep, grazing in a paddock, is tethered to a stake by a rope 20 m long. If the
stake is 10 m from a fence, find the area over which the sheep can graze.
16. Two tangents, PA and PB, are drawn to a circle with centre 0 and radius 5 cm
from an external point, P. If PO = 13 cm, calculate the area hounded by the
tangents and the minor arc AB.
II.
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