Circle Theorems 1
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Rayner: Intermediate GCSE Mathematics Revision and Practice
Circle theorems
Circle theorems There are four theorems on angles in circles that you should know. First you need to learn some words.
A D
The straight line AB is a chord. The curved line AB (in bold) is an arc. The chord AB divides the circle into two segments ± the major segment (shaded) and the minor segment (unshaded). b is the angle subtended by AB at C. ACB ABCD is a cyclic quadrilateral.
l l l l l
Theorem 1 The angle subtended at the centre of a circle is twice the angle subtended at the circumference.
B
C
Theorem 2 Angles subtended by an arc in the same segment of a circle are equal. Y
C
Z
x X O 2x A
B
A
b 2 ACB b AOB
B
b AYB b AZB b AXB
The proof of this theorem is given in the section on proof. b b 448 ®nd BAC b 628 and DCA (b) Given BDC b and ABD:
b b 508, ®nd BCA: (a) Given ABO
D B 62° 50° O
44°
C
C
A A B
Triangle OBA is isosceles (OA OB:
b (both subtended by arc BC) b BAC BDC b ; BAC 62 b ABD b (both subtended by arc DA) DCA b 44 ; ABD
b 50 ; OAB b 80 (angle sum of a triangle) ; BOA b 40 (angle at the centre) ; BCA
Exercise 1 Find the angles marked with letters. A line passes through the centre only when point O is shown.
1.
2. 27°
3. 20°
a
d 45°
4. 58°
30°
f d
g 85°
41°
c
b
c 30°
e
40°
h
1
Rayner: Intermediate GCSE Mathematics Revision and Practice
5.
Circle theorems
7.
6. 32°
25°
96° c
c
c
8.
B
t
46° O
a 80°
40° y
50°
b
a
94°
C
A
10.
9.
12.
11.
A
A
B
c
C
B
38° A
e 84°
O
b
d
B
O
88°
h
O
O
g
98°
C
f
C
A
C B
14.
13.
15.
C
B
D
16.
C x
y
a
58°
42°
108°
O
O
O A
A x A
x ⫹ 22°
x
B
B
C
Theorem 3 The opposite angles in a cyclic quadrilateral add up to 1808 (the angles are supplementary).
Find a and x. a 180 81
C
(opposite angles of a cyclic quadrilateral)
D
; a 99
A
b C b 1808 A b b B D 1808
Theorem 4 The angle in a semicircle is a right angle.
B
2x D
81°
a
x 2x 180 (opposite angles of a cyclic quadrilateral)
C
B
O
x A
3x 180 ; x 60
Find b given that AOB is a diameter.
A A 37°
O
O
b C
B
B
C
In the diagram, AB is a diameter. b 908. ACB
b 90 (angle in a semicircle) ACB ; b 180 90 37 53 2
Rayner: Intermediate GCSE Mathematics Revision and Practice
Circle theorems
Exercise 2 Find the angles marked with a letter.
1.
2.
3.
4. x
101°
b
2c
d
116°
96°
3d
86°
c 105°
c
a
y
d 92°
5.
E
6. C 106°
B
7.
E
8.
B A
e
D m
h
A
B
9.
A
10.
a
B
72°
B
15.
n
b
t
O
A
a
18°
16.
C
D
C
50°
B
w
O
O
3x
14. D 32°
x
B
C
B
A
A
A
m
13.
12. 2a
O
D
E
C
33°
O
y D
11.
A
A x
4e
C
D
C
f
A
31°
C
140° B
O
x
c
C
B
c a
B
O
A
4a
O
35°
A
O
C B
D
17.
18.
D
A
g 82° e A
19.
O
C
40° B
z
C
20.
C
B 95°
O
y
O f B
32° D
x
C
z
120° x A
50° B
3
Rayner: Intermediate GCSE Mathematics Revision and Practice
Moving averages
Moving averages Here is a list of the number of children absent from a school over a 15-day period. Day
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
Number absent
10
7
4
12
6
3
5
8
9
11
13
8
10
7
5
At the end of each day the mean number absent over the last ®ve days is calculated. So at the end of day 5, the mean over the last ®ve days is
10 7 4 12 6 78 5
At the end of day 6, the mean over the last ®ve days is
7 4 12 6 3 64 5
At the end of day 7, the mean over the last ®ve days is
4 12 6 3 5 6 5
This is an example of a moving average.
Exercise 3 1. Traders on the stock market use moving averages as a guide to the performance of a company's share price. Here are the share prices, in pence, of a company over 30 days. 21 28 22
24 27 21
27 28 19
22 26 19
25 25 20
26 23 19
27 25 21
23 26 23
24 23 24
24 25 23
(a) What was the mean price over the ®rst 10 days? (b) What was the mean price over the 10 day period from day 2 to day 11? (c) What was the mean price over the 10-day period from day 11 to day 20? 2. Here are the prices, in pence, of shares in `Tiger Telecom' over a period of 20 days. 8 15
7 14
11 16
10 15
9 13
7 16
9 14
11 10
14 12
15 13
At the end of each day the mean price over the last 5 days is calculated. 8 7 11 10 9 So the mean price at the end of day 5 is 9p 5 The mean price on day 6
7 11 10 9 7 88p 5
Work out the moving average price of the shares in this way up to day 20 and plot the results on a graph.
Moving average price (p) 9 8 7
5
6
7
8
Day
4
Rayner: Intermediate GCSE Mathematics Revision and Practice
Answers
Answers Exercise 1 1. 4. 7. 10. 13. 16.
a 278, b 308 f 408, g 558, h 558 438 c 468, d 448 488 a 368, x 368
2. 5. 8. 11. 14.
c 208, d 458 a 328, b 808, c 438 928 e 498, f 418 328
3. 6. 9. 12. 15.
c 588, d 418, e 308 c 348, y 348 428 g 768, h 528 228
2. 5. 8. 11. 14. 17. 20.
c 1018, d 848 378 358 308 a 328, b 408, c 408 e 418, f 418, g 418 x 808, z 108
3. 6. 9. 12. 15. 18.
x 928, y 1168 1188 188 22 12 8 a 188, c 728 88
Exercise 2 1. 4. 7. 10. 13. 16. 19.
a 948, b 758 c 608, d 458 e 368, f 728 908 n 588, t 648, w 458 558 x 308, y 1158
Exercise 3 1. (a) 243p (b) 25p (c) 256p 2. averages: 9, 88, 92, 92, 10, 112, 128, 138, 148, 15, 146, 148, 148, 136, 13, 13
5
Circle theorems
Circle theorems There are four theorems on angles in circles that you should know. First you need to learn some words.
A D
The straight line AB is a chord. The curved line AB (in bold) is an arc. The chord AB divides the circle into two segments ± the major segment (shaded) and the minor segment (unshaded). b is the angle subtended by AB at C. ACB ABCD is a cyclic quadrilateral.
l l l l l
Theorem 1 The angle subtended at the centre of a circle is twice the angle subtended at the circumference.
B
C
Theorem 2 Angles subtended by an arc in the same segment of a circle are equal. Y
C
Z
x X O 2x A
B
A
b 2 ACB b AOB
B
b AYB b AZB b AXB
The proof of this theorem is given in the section on proof. b b 448 ®nd BAC b 628 and DCA (b) Given BDC b and ABD:
b b 508, ®nd BCA: (a) Given ABO
D B 62° 50° O
44°
C
C
A A B
Triangle OBA is isosceles (OA OB:
b (both subtended by arc BC) b BAC BDC b ; BAC 62 b ABD b (both subtended by arc DA) DCA b 44 ; ABD
b 50 ; OAB b 80 (angle sum of a triangle) ; BOA b 40 (angle at the centre) ; BCA
Exercise 1 Find the angles marked with letters. A line passes through the centre only when point O is shown.
1.
2. 27°
3. 20°
a
d 45°
4. 58°
30°
f d
g 85°
41°
c
b
c 30°
e
40°
h
1
Rayner: Intermediate GCSE Mathematics Revision and Practice
5.
Circle theorems
7.
6. 32°
25°
96° c
c
c
8.
B
t
46° O
a 80°
40° y
50°
b
a
94°
C
A
10.
9.
12.
11.
A
A
B
c
C
B
38° A
e 84°
O
b
d
B
O
88°
h
O
O
g
98°
C
f
C
A
C B
14.
13.
15.
C
B
D
16.
C x
y
a
58°
42°
108°
O
O
O A
A x A
x ⫹ 22°
x
B
B
C
Theorem 3 The opposite angles in a cyclic quadrilateral add up to 1808 (the angles are supplementary).
Find a and x. a 180 81
C
(opposite angles of a cyclic quadrilateral)
D
; a 99
A
b C b 1808 A b b B D 1808
Theorem 4 The angle in a semicircle is a right angle.
B
2x D
81°
a
x 2x 180 (opposite angles of a cyclic quadrilateral)
C
B
O
x A
3x 180 ; x 60
Find b given that AOB is a diameter.
A A 37°
O
O
b C
B
B
C
In the diagram, AB is a diameter. b 908. ACB
b 90 (angle in a semicircle) ACB ; b 180 90 37 53 2
Rayner: Intermediate GCSE Mathematics Revision and Practice
Circle theorems
Exercise 2 Find the angles marked with a letter.
1.
2.
3.
4. x
101°
b
2c
d
116°
96°
3d
86°
c 105°
c
a
y
d 92°
5.
E
6. C 106°
B
7.
E
8.
B A
e
D m
h
A
B
9.
A
10.
a
B
72°
B
15.
n
b
t
O
A
a
18°
16.
C
D
C
50°
B
w
O
O
3x
14. D 32°
x
B
C
B
A
A
A
m
13.
12. 2a
O
D
E
C
33°
O
y D
11.
A
A x
4e
C
D
C
f
A
31°
C
140° B
O
x
c
C
B
c a
B
O
A
4a
O
35°
A
O
C B
D
17.
18.
D
A
g 82° e A
19.
O
C
40° B
z
C
20.
C
B 95°
O
y
O f B
32° D
x
C
z
120° x A
50° B
3
Rayner: Intermediate GCSE Mathematics Revision and Practice
Moving averages
Moving averages Here is a list of the number of children absent from a school over a 15-day period. Day
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
Number absent
10
7
4
12
6
3
5
8
9
11
13
8
10
7
5
At the end of each day the mean number absent over the last ®ve days is calculated. So at the end of day 5, the mean over the last ®ve days is
10 7 4 12 6 78 5
At the end of day 6, the mean over the last ®ve days is
7 4 12 6 3 64 5
At the end of day 7, the mean over the last ®ve days is
4 12 6 3 5 6 5
This is an example of a moving average.
Exercise 3 1. Traders on the stock market use moving averages as a guide to the performance of a company's share price. Here are the share prices, in pence, of a company over 30 days. 21 28 22
24 27 21
27 28 19
22 26 19
25 25 20
26 23 19
27 25 21
23 26 23
24 23 24
24 25 23
(a) What was the mean price over the ®rst 10 days? (b) What was the mean price over the 10 day period from day 2 to day 11? (c) What was the mean price over the 10-day period from day 11 to day 20? 2. Here are the prices, in pence, of shares in `Tiger Telecom' over a period of 20 days. 8 15
7 14
11 16
10 15
9 13
7 16
9 14
11 10
14 12
15 13
At the end of each day the mean price over the last 5 days is calculated. 8 7 11 10 9 So the mean price at the end of day 5 is 9p 5 The mean price on day 6
7 11 10 9 7 88p 5
Work out the moving average price of the shares in this way up to day 20 and plot the results on a graph.
Moving average price (p) 9 8 7
5
6
7
8
Day
4
Rayner: Intermediate GCSE Mathematics Revision and Practice
Answers
Answers Exercise 1 1. 4. 7. 10. 13. 16.
a 278, b 308 f 408, g 558, h 558 438 c 468, d 448 488 a 368, x 368
2. 5. 8. 11. 14.
c 208, d 458 a 328, b 808, c 438 928 e 498, f 418 328
3. 6. 9. 12. 15.
c 588, d 418, e 308 c 348, y 348 428 g 768, h 528 228
2. 5. 8. 11. 14. 17. 20.
c 1018, d 848 378 358 308 a 328, b 408, c 408 e 418, f 418, g 418 x 808, z 108
3. 6. 9. 12. 15. 18.
x 928, y 1168 1188 188 22 12 8 a 188, c 728 88
Exercise 2 1. 4. 7. 10. 13. 16. 19.
a 948, b 758 c 608, d 458 e 368, f 728 908 n 588, t 648, w 458 558 x 308, y 1158
Exercise 3 1. (a) 243p (b) 25p (c) 256p 2. averages: 9, 88, 92, 92, 10, 112, 128, 138, 148, 15, 146, 148, 148, 136, 13, 13
5
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