The document provides information about calculating geometric properties of circles such as circumference, area, perimeter and area of sectors, as well as the volumes of cylinders. It includes formulas, examples of calculations, and multiple choice questions to test understanding. Key formulas covered include circumference = π x diameter, area of a circle = π x radius^2, area of a sector = (angle of sector/360) x π x radius^2, and volume of a cylinder = area of base x height. The document provides a review of foundational circle geometry concepts through examples, practice questions, and matching definitions to geometric terms.

1. Circles

2. Pi
Circle words
Rounding
Refresher
Area
Perimeter and
Area of
compound shapes
Perimeters of
sectors
Area of Sectors
Volumes of
Cylinders
Volume of
Spheres and
cones
Equation of a
circle 2
Simultaneous
Equations
Finding the
radius of
sectors
Radius and
Height of
Cylinders
Circle theorems
Area of
Segments
Equation of a
circle 1
Circle formulae
Circumference

3. Match the words to the definitions
•Sector
•Segment
•Chord
•Radius
•Arc
•Tangent
•Diameter
•Circumference
•The length around the outside of a circle
•A line which just touches a circle at one
point
•A section of a circle which looks like a
slice of pizza
•A section circle formed with an arc and a
chord
•The distance from the centre of a circle to
the edge
•The distance from one side of a circle to
the other (through the centre)
•A section of the curved surface of a circle
•A straight line connecting two points on
the edge of a circle
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4. Think about circles
Image that line straightened out- this outside of a circle
Think about a line around theis the circumference

5. Pi
People noticed that if you divide the circumference of a circle by the
diameter you ALWAYS get the same answer
They called the answer Pi (π) , which is:
3.1415926535897932384626433832795028841971693993751
058209749445923078164062862089986280348253421170679
821480865132823066470938446095505822317253594081284
You can use the
8111745028410270193852110555964462294895493038196
π button on your
442881097566593344612847564823378678316527120190914
calculator
5648566923460348610454326648213393607260249141273
724587006606315588174881520920962829254091715364367
892590360011330530548820466521384146951941511609...

6. How many digits can you memorise in
2 minutes?
3.141592653589793238462643383279502884197169399375
10582097494459230781640628620899862803482534211
70679821480865132823066470938446095505822317253
59408128481117450284102701938521105559644622948
95493038196442881097566593344612847564823378678
31652712019091456485669234603486104543266482133
93607260249141273724587006606315588174881520920
96282925409171536436789259036001133053054882046
6521384146951941511609...

7. Write down pi!
3.141592653589793238462643383279502884197169399375
10582097494459230781640628620899862803482534211
70679821480865132823066470938446095505822317253
59408128481117450284102701938521105559644622948
95493038196442881097566593344612847564823378678
31652712019091456485669234603486104543266482133
93607260249141273724587006606315588174881520920
96282925409171536436789259036001133053054882046
6521384146951941511609...
How did you do? What do you think
the world record is?

8. Pi website

9. Pi Story
• One way to memorise Pi is to write a Pi-em (pi
poem) where the number of letters in each word
is the same as the number in pi. For example:
“Now I, even I, would celebrate in rhymes inept,
the great immortal Syracusan rivall'd nevermore
who in his wondrous lore passed on before
left men his guidance how to circles mensurate.”
Can you write one of your own?
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10. Rounding to Decimal Places
10 multiple choice
questions

11. Round to
1 dp
0.34
A)
0.4
B)
0.3
C)
0.35
D)
3

12. Round to
1 dp
0.48
A)
0.49
B)
0.4
C)
0.47
D)
0.5

13. Round to
1 dp
2.75
A)
2.8
B)
2.74
C)
2.7
D)
3.0

14. Round to
1 dp
13.374
A)
13.0
B)
14.0
C)
13.3
D)
13.4

15. Round to
1 dp
26.519
A)
25.0
B)
26.6
C)
26.0
D)
26.5

16. Round to
2 dp
23.1782
A)
23.20
B)
23.18
C)
23.10
D)
23.17

17. Round to
2 dp
500.8251
A)
500.80
B)
500.8
C)
500.83
D)
500.84

18. Round to
3 dp
0.00417
A)
0.00417
B)
0.004
C)
0.00418
D)
0.005

19. Round to
2 dp
4.999
A)
4.99
B)
4.90
C)
5.00
D)
4.98

20. Round to
4 dp
0.72995
A)
0.7300
B)
0.7390
C)
0.7210
D)
0.7399
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21. Finding the Circumference
You can find the circumference of a circle by using the formula-
Circumference = π x diameter
For Example10cm
Area= π x 10
= 31.41592654....
= 31.4 cm (to 1 dp)

22. You can find the circumference of a circle by using the formula-
Circumference = π x diameter
For Example-
10cm
Area= π x 10
= 31.41592654....
= 31.4 cm (to 1 dp)
Find the Circumference of a circles with:
1. A diameter of :
a) 8cm
b) 4cm
c) 11cm
d) 21cm
e) 15cm
2. A radius of :
a) 6cm
b) 32cm
c) 18cm
d) 24cm
e) 50cm
ANSWERS
1a 25.1cm
b 12.6cm
c 34.6cm
d 66.0cm
e 47.1cm
2a
b
c
d
e
37.7cm
201.1cm
113.1cm
150.8cm
157.1cm
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23. Finding the Area
You can find the area of a circle by using the formula-
Area= π x Radius2
For Example7cm
Area= π x 72
= π x 49
= 153.93804
= 153.9 (to 1dp) cm2

24. Finding the Area
You can find the area of a circle by using the formula-
Area= π x Radius2
7cm
For ExampleArea= π x 72
= π x 49
= 153.93804
= 153.9 (to 1dp) cm2
ANSWERS
2a
b
c
d
e
f
g
h
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12.6
78.5
15.2
380.1
314.2
153.9
100.5
28.3

25. Finding the Area of a Sector
To find the area of a sector, you need to work out what fraction of a full
circle you have, then work out the area of the full circle and find the
fraction of that area.
For ExampleThe sector here is ¾ of a full circle
Find the area of the full circle
7cm
Area= π x 72
= π x 49
= 153.93804
= DON’T ROUND YET!
Then find ¾ of that area
¾ of 153.93804 = 115.45353 (divide by
4 and multiply by 3)

26. Finding the Area of a Sector
Sometimes it is not easy to see what fraction of a full circle you have.
You can work it out based on the size of the angle. If a full circle is 360°
, and this sector is 216°, the sector is 216/360, which can be simplified
to 3/5.
For Example-
The sector here is 3/5 of a full circle
Find the area of the full circle
216°
7cm
Area= π x 72
= π x 49
= 153.93804
= DON’T ROUND YET!
Then find 3/5 of that area
3/5 of 153.93804 = 92.362824 (divide
by 5 and multiply by 3)
= 92.4cm2
Sometimes the fraction cannot be simplified and will stay over 360

27. Finding the Area of a Sector
The general formula for finding the area is:
Area of sector= Angle of Sector x πr2
360
Fraction of full
circle that sector
covers
“of”
Area of full
circle

28. Questions
Find the area of these sectors, to 1 decimal place
1
3
2
10cm
11cm
260°
190°
4
12cm
251°
6
5
87°
ANSWERS
5cm
6.5cm
32°
1
2
3
4
5
6
226.9
200.6
315.4
19.0
61.2
80.7
17cm
166°
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29. Finding the Perimeter of a
Sector
To find the perimeter of a sector, you need to work out what fraction of a full
circle you have, then work out the circumference of the full circle and find the
fraction of that circumference.
You then need to add on the radius twice, as so far you have worked out the length
of the curved edge
For Example-
The sector here is ¾ of a full circle
Find the area of the full circle
7cm
Area= π x 14 (the diameter is twice the radius)
= π x 49
= 43.982297......
= DON’T ROUND YET!
Then find ¾ of that circumference
¾ of 43.982297...... = 32.99 cm (2 dp)
Remember to add on 7 twice from the straight
sides

30. Finding the Area of a Sector
Sometimes you will not be able to see easily what fraction of the full
circle you have.
To find the fraction you put the angle of the sector over 360
250°
This sector is 250/360 or two
hundred and fifty, three
hundred and sixty-ITHS of the
full circle
Simplify if you can
Sometimes the fraction cannot be simplified and will stay over 360

31. Finding the Perimeter of a
Sector
The general formula for finding the area is:
Perimeter of sector= (Angle of Sector x πd) + r + r
360
Fraction of full circle
that sector covers
“of”
Circumference of full
circle
Don’t forget the
straight sides
This is the same as d
of 2r, but I like r +r as
it helps me
remember why we
do it

32. Questions
Find the perimeter of these sectors, to 1 decimal place
1
3
2
10cm
11cm
260°
190°
4
251°
6
5
87°
12cm
5cm
6.5cm
32°
ANSWERS
1
65.4
2
58.5
3
76.6
4
17.6
5
31.8
6
43.5
17cm
166°
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33. Compound Area and Perimeter
Here we will look at shapes made up of triangles, rectangles, semi and quarter circles.
Find the area of the shape below:
10cm
8cm
10cm
Area of this
rectangle= 8 x10
=80cm2
Area of this semi circle = π r2 ÷ 2
= π x 52 ÷ 2
= π x 25 ÷ 2
=39.3 cm2 (1dp)
Area of whole shape = 80 + 39.3
= 119.3 cm2

34. Compound Area and Perimeter
Find the perimeter of the shape below:
10cm
8cm
10cm
Perimeter of this
rectangle= 8 + 8 + 10
=26cm
(don’t include the red
side)
Circumference of this semi circle = πd ÷ 2
= π x 10 ÷ 2
=15.7 cm (1dp)
Perimeter of whole shape = 26 +
15.7
= 31.7 cm

35. Compound Area and Perimeter
Find the areaof the shape below:
Area of this quarter circle = π r2 ÷ 4
= π x 52 ÷ 4
= π x 25 ÷ 4
=19.7 cm2 (1dp)
5cm
10cm
11cm
Area of this rectangle
10 x 11=110
Area of whole shape = 110+ 19.7
= 129.7cm2

36. Compound Area and Perimeter
Work out all missing
Find the perimeter of the shape below: sides first
?
5cm
5cm
Circumference of this quarter circle = πd ÷ 4
= π x 10 ÷ 4 (if radius is 5, diameter is 10)
=7.9 cm (1dp)
6cm
10cm
10cm
11cm
Add all the straight
sides=
10+10 + 11+ 5 + 6= 42cm
Area of whole shape = 42+ 7.9
= 49.9cm

37. Questions
ANSWERS
AREA PERIMETER
1 38.1
23.4
2 135.0
61.3
Find the perimeter and area of these shapes, to 1 decimal place
3 181.1
60.8
2cm
3 4 27.3
1
2
12cm
10cm
5 129.3
47.7
20cm
6 128.5
6cm
11cm
4cm
17cm
4cm
4
Do not worry about
perimeter here
5cm
12cm
6
6cm
5
Do not worry about
perimeter here
10cm
5cm
5cm
10cm
20cm
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38. Volume of Cylinders
Here we will find the volume
of cylinders
Cylinders are prisms with a
circular cross sections, there
are two steps to find the
volume
1) Find the area of the circle
1) Multiple the area of the
circle by the height or length
of the cylinder

39. Volume of Cylinders 2
EXAMPLE- find the volume of
this cylinder
4cm
1) Find the area of the circle
π x r2
π x 42
π x 16 = 50.3 cm2 (1dp)
10cm
2) Multiple the area of the
circle by the height or length
of the cylinder
50.3 (use unrounded answer
from calculator) x 10 = 503cm3

40. Questions
Find the volume of these cylinders, to 1 decimal place
1
2
4cm
3cm
3
ANSWERS
5cm
10cm
15cm
603.2
2
12cm
1
282.7
3 1178.1
4 142.0
5 2155.1
4
2cm
5
11.3cm
6
7cm
14cm
6
508.9
3cm
18cm
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41. Volume of Cylinders 2
EXAMPLE- find the height of
this cylinder
4cm
1) Find the area of the circle
π x r2
π x 42
π x 16 = 50.3 cm2 (1dp)
h
2) Multiple the area of the
circle by the height or length
of the cylinder
50.3 x h = 140cm3
Rearrange this to give
h= 140 ÷ 50.3
h=2.8 cm
Volume=
140cm3

42. Volume of Cylinders
EXAMPLE- find the radius of
this cylinder
r
1) Find the area of the circle
π x r2
2) Multiple the area of the
circle by the height or length
of the cylinder
π x r2 x 30 = 250cm3
94.2... x r2 = 250
Rearrange this to give
r2 = 250 ÷ 94.2
r2 =2.7 (1dp)
r= 1.6 (1dp) cm
30cm
Volume=
250cm3

43. Questions
1
ANSWERS
1 6.4
2 4.2
3
Find the volume of these cylinders, to 1 decimal place 1.3
4 2.3
3
4cm
2
5 1.8
3cm
5cm
6 1.9
h
h
volume= 320cm3
4
volume= 120cm3
5
r
volume= 100cm3
6
r
12cm
volume= 200cm3
h
r
8cm
14cm
volume= 150cm3
volume= 90cm3
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44. Volume of Spheres
The formula for the volume of
a sphere is
e.g
A= 4/3 x π x 103
A= 4/3 x π x 1000
A=4188.8 cm3 (1 dp)
10cm

45. Volume of Cones
The formula for the volume of
a cone is
e.g
A= 1/3 x π x 42 x 10
A= 1/3 x π x 16 x 10
A=167.6 cm3 (1 dp)
10cm
4cm

46. Questions
1
ANSWERS
1 4188.8
2 33510.3
3
Find the volume of these spheres, to 1 decimal place 523.6
4 201.1
3
5 122.5
2
6 1272.3
10cm
4
20cm
5cm
6
5
12cm
13cm
4cm
3cm
15cm
9cm
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47. Circles
Theorems
Angle at the
centre
Angles
connected by a
chord
Triangles made
with a diameter
or radii
Cyclic
Quadrilaterals
Tangents

48. Double Angle
The angle at the centre of a circle is twice the angle at the
edge
Example
Angle x = 50 x 2
x=100°
50°
x

49. 1
Answers
1) 50
2)120
3)180
4)50
5)67.5
6)80
4
2
3
25°
60°
90°
x
x
x
x
100°
5
x
135°
6
x
160°
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50. Triangles inside circles
A triangle containing a diameter, will be a right angled triangle
A triangle containing two radii, will be isosceles
90°
x
x

51. 2
1
3
72°
x
x
60°
Answers x
1) X=30
2)x=18
3)x=45
4)X=40 y=40
5)x=30 y= 120
6)x=22 y=136
1
x
x
x
y
3
2
x
22°
100°
y
30°
y
x
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52. Angles connected by a chord
Angles connected by a chord are equal
x
x
y
y

53. 2
1
25°
x
15°
Answers
1) x=25 y=15
2)x=125 y= 40 z=15
3)x=10 y=70 z=100
4)X=105 y=40 z=35
5)x=53 y= 30 z=72
4
6)x=85 y=80 z=17
y
x
10°
x
40°
x
y
z
3
y
z 100°
125°
z
15°
y
5
6
z
y
40°
35°
x
25°
30°
53°
y
x
17°
80°
95°
z
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54. Tangents to a circle
A tangent will always meet a radius at 90°
90°

55. 1
2
x
35 °
x
140°
4
z
3
x
40°
z
120°
x
y
y
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56. Cyclic Quadrilaterals
Opposite angles in a cyclic quadrilaterals add up to 180°
100°
60°
x
y
100 + y = 180
y=80°
60 + x = 180
x = 120 °

57. 2
1
95°
3
54°
110°
20°
x
y
x
y
Answers
1) x=70 y=85
2)x=126 y=105
3)x=100 y=160
4)w=15 x=70
y=65 z= 25
5)a=60 b=36
x
75°
4
80°
y
5
15°
25°
70°
a
w
4b
y
z
x
b
2a
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58. Area of Segments
Here we will look at finding the area of sectors
You will need to be able to do two things:
1) Find the area of a sector using the
formula-
2) Find the area of a triangle using
the formulaArea= ½ absinC
Area of sector= Angle of Sector x πr2
360
a
C
b

59. Examplefind the area of the blue segment
Step 1- find the area of the whole sector
Area= 100/360 x π x r2
= 100/360 x π x 102
=100/360 x π x 100
=87.3cm2
10cm
100°
10cm
Step 2- find the area of the triangle
Area= ½ absinC
=1/2 x 10 x 10 x sin100
= 49.2cm2
Step 3- take the area of the triangle from
the area of the segment
87.3 – 49.3 = 38 cm2

60. Examplefind the area of the blue segment
Step 1- find the area of the whole sector
Area= 120/360 x π x r2
= 120/360 x π x 122
=120/360 x π x 144
=150.8cm2
12cm
120°
12cm
Step 2- find the area of the triangle
Area= ½ absinC
=1/2 x 12 x 12 x sin120
= 62.4cm2
Step 3- take the area of the triangle from
the area of the segment
150.8 – 62.4 = 88.4 cm2

61. Questions
1
ANSWERS
1 75.1
2 29.5
3
Find the area of the blue segments, to 1 decimal place 201.1
4 8.3
3
2
5 51.8
10cm
6 33.0
85° 11cm
170° 12cm
130°
4
6
5
95°
6.5cm
5cm
160°
65°
17cm
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62. Finding the Radius or angle of a
Sector
r
100°
Area=200cm2
Area= 100 x π x r2
360
200= 100 x π x r2
360
200x360 = r2
100 x π
229.2=r2
15.1cm =r
10cm
x
Area=150
Area= θ x π x r2
360
150= θ x π x 102
360
150x360 = θ
102 x π
117.9°= θ

63. Questions
ANSWERS
1 7.6
2 8.3
Find the missing radii and angles of these sectors, to 1 decimal 3 5.4
place
4 160.4
3
1
2
5 122.1
r
r
r
6 47.6
200°
175°
Area=100cm2
4
250°
Area=120cm2
6
5
5cm
θ
Area=35cm2
6.5cm
Area=50cm2
17cm
θ
Area=120cm2
θ
Area=45cm2
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64. The Equation of a Circle
The general equation for a circle is (x-a)2 + (y-b)2=r2
This equation will give a
circle whose centre is at
(a,b) and has a radius of r
For example a circle has the equation (x-2)2 + (y-3)2=52
This equation will give a
circle whose centre is at
(2,3) and has a radius of 5

65. The Equation of a Circle
A circle has the equation (x-5)2 + (y-7)2=16
This equation will give a circle whose centre
is at (5,7) and has a radius of 4 (square root
of 16 is 4)
For example a circle has the equation (x+2)2 + (y-4)2=100
This equation will give a circle
whose centre is at (-2,4) and has a
radius of 10
You could think of
this as (x - -2)2

66. The Equation of a Circle
A circle has the equation (x-5)2 + (y-7)2=16
What is y when x is 1?
(1-5)2 + (y-7)2=16
12+ (y-7)2=16
1+ (y-7)2=16
(y-7)2=15
y-7= ±3.9 (square root of 15 to 1 dp)
y= 7±3.9
y= 10.9 or 3.1
There are two coordinates on the circle with x=1, one is
(1,10.9) and the other is (1,3.1)

67. The Equation of a Circle
1) Write down the coordinates of the centre point and radius of each of these circles:
Answers
a)
(x-5)2 + (y-7)2=16
b)
(x-3)2 + (y-8)2=36
c)
(x+2)2 + (y-5)2=100
d)
(x+2)2
e)
(x-6)2 + (y+4)2=144
f)
x2 + y2=4
g)
x2 + (y+4)2=121
h)
(x-1)2 + (y+14)2 -16=0
i)
1a) r=4 centre (5,7)
b) r=6 centre (3,8)
c) r=4 centre (-2,5)
d) r=10 centre (-2,-5)
e) r=7 centre (6,-4)
f) r=12 centre (0,0)
g) r=411centre (0,-4)
h) r=4 centre (1,-14)
i) r=5 centre (5,9)
(x-5)2 + (y-9)2 -10=15
+
(y+5)2=49
2) What is the diameter of a circle with the equation (x-1)2 + (y+3)2 =64
Answers
2) 16
3)Circumference = 25.1
Area=50.3
4)Circumference = 25.1
Area=50.3
5) Circles have the same
radius but different centres,
they are translations
3) Calculate the area and circumference of the circle with the equation (x-5)2 + (y-7)2=16
4) Calculate the area and perimeter of the circle with the equation (x-3)2 + (y-5)2=16
5) Compare your answers to question 3 and 4, what do you notice, can you explain this?
6 ) A circle has the equation (x+2)2 + (y-4)2=100, find:
a) x when y=7
b) y when x=6
6a) x= 11.5 or -7.5
b) y=11.3 or -3.3
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68. The Equation of a Circle 2
Here we will look at rearranging equations to find properties of the circle they
represent
Remember- The general equation for a circle is (x-a)2 + (y-b)2=r2
The skill you will need is called completing the
square, you may have used it to solve
quadratic equations

69. The Equation of a Circle 2
Example
x2 + y2 -6x – 8y =0
Create two brackets and put x in one and y in the other
(x
) 2 + (y
)2 = 0
Half the coefficients of x and y and put them into the brackets, and
then subtract those numbers squared
(x -3) 2 + (y - 4) 2 – 32 - 42= 0
Tidy this up
(x -3) 2 + (y - 4) 2 – 25= 0
(x -3) 2 + (y - 4) 2 = 25
This circle has a
radius of 5 and
centre of (3,4)

70. The Equation of a Circle 2
Example
x2 + y2 -10x – 4y- 7 =0
Create two brackets and put x in one and y in the other
(x
) 2 + (y
)2 = 0
Half the coefficients of x and y and put them into the brackets, and
then subtract those numbers squared
(x -5) 2 + (y - 2) 2 – 52 – 22 - 7= 0
Tidy this up
(x -5) 2 + (y - 2) 2 – 36= 0
(x -5) 2 + (y - 2) 2 = 36
This circle has a
radius of 6 and
centre of (5,2)

71. The Equation of a Circle 2
You must always make sure the coefficient of x2 and y2 is 1
You may have to divide through 2x2 + 2y2 -20x – 8y- 14 =0
Divide by 2 to give x2 + y2 -10x – 4y- 7 =0
Then put into the form x2 + y2 -10x – 4y- 7 =0

72. Questions
Put this equations into the form (x-a)2 + (y-b)2=r2 then find
the centre and radius of the circle
1.
2.
3.
4.
5.
6.
7.
8.
x2 + y2 -8x – 4y- 5 =0
x2 + y2 -12x – 6y- 4 =0
x2 + y2 -4x – 10y- 20 =0
x2 + y2 -10x – 14y- 7 =0
x2 + y2 -12x – 2y- 62 =0
2x2 +2y2 -20x – 20y- 28 =0
3x2 + 3y2 -42x – 24y- 36 =0
5x2 + 5y2 -100x – 30y- 60 =0
Answers
1) r=5 centre (4,2)
2) r=7 centre (6,3)
3) r=7 centre (2,5)
4) r=9 centre (5,7)
Answers
5) r=10 centre (6,1)
6) r=8 centre (5,5)
7) r=8 centre (6,4)
8) r=11 centre (10,3)
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73. Simultaneous Equations
A circle has the equation (x-5)2 + (y-7)2=16 and a line has an
equation of y=2x+1, at what points does the line intercept
the circle?
We need to substitute into the equation of the circle so that we only have x’s or y’s
Because y=2x +1 we can rewrite the equation of the circle but instead of putting “y”
in we’ll write “2x+1”
Ways to solve quadratic
equationsCompleting the square
Factorising
The Quadratic formula
So,
(x-5)2 + (2x-1-7)2=16
(x-5)2 + (2x-8)2=16
expand the brackets
x2-10x + 25 + 4x2 – 32x +64 = 16 simplify and make one side 0
5x2 -42x + 73=0 solve this quadratic equation to find x,
Put the value / values of x into y=2x+1 to find the coordinates of the intercept /
intercepts to answer the question

74. Simultaneous Equations
A quadratic equation can give 1,2 or no solutions, a line can
cross a circle at 1,2 or no points
1 solution to the
quadraticThe line is a tangent
2 solutions to the
quadratic
0 solutions to the
quadratic the circle
and the line never
meet

75. Intercepts between lines and circles
1) Find out whether these circles and lines intercept, if they do find the coordinates
of the interceptions
a) (x-5)2 + (y-7)2=16
and
b) (x-3)2 + (y-8)2=36
and
c) (x+2)2 + (y-5)2=100
and
d) (x+2)2 + (y+5)2=49
and
e) (x-6)2 + (y+4)2=144
and
ANSWERS (all have been
rounded)
y=2x-2
(3.6,9.8) and (2.2,5.6)
y=3x + 3
(7.2,12.3) and (2,2)
2y+4=x
(3.5,13.4) and (-2.7,-5)
y -3x =5 (-0.4,3.3) and (-6.4,-8.9)
(-4.6,6.9) and (-4.6,-8.9)
y=3x-1
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76. Circle Formulae
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