Circles

1
Circles
General
Isosceles triangles in a circle
Angles in a semi-circle
Pythagoras Theorem
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SOHCAHTOA
Tangent line on a circle
2
Starter Questions
General
B
6
co
C
A
8
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3
Isosceles triangles in Circles
General
Aim of Todays Lesson
To identify isosceles triangles within a circle.
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4
Isosceles triangles in Circles
General
When a line is drawn between two points on a
circle it is called a CHORD.
If the line passes through the Centre it is
called a diameter.
A
B
xo
xo
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When two radii are drawn to the ends of a chord,
an isosceles triangle is formed.
C
5
Isosceles triangles in Circles
General
Special Properties of Isosceles Triangles
Two equal lengths
Two equal angles
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Angles in any triangle sum to 180o
6
Isosceles triangles in Circles
General
Q. Find the angle xo.
B
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C
xo
Since the triangle is isosceles we have
A
280o
7
Isosceles triangles in Circles
General
Begin Maths in Action Worksheet 1 or Ex 2
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8
Starter Questions
General
B
bo
5
C
A
12
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27o
9
Angles in a Semi-Circle
General
Aim of Todays Lesson
To find the special angle in a semi-circle
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10
Angles in a Semi-Circle
General
On your worksheet, mark the point P on the
circumference of the semicircle.
Joint the points AP and BP together. Measure
angle ?APB
P
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A
B
The angle in a semi-circle is ALWAYS 90o
11
Angles in a Semi-Circle
General
KeyPoint for Angles in a Semi-circle
A triangle APB inscribed within a semicircle
with hypotenuse equal to the diameter will
ALWAYS be right angled at P on the circumference.
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Remember - Angles in any triangle sum to 180o
12
Angles in a Semi-Circle
General
Example 1 Sketch diagram and find all the
missing angles.
20o
43o
Look for right angle triangles
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Remember ! Angles in any triangle sum to 180o
47o
70o
13
Angles in a Semi-Circle
General
Example 2 Sketch the diagram.
(a) Right down two right angle triangles
(a) Calculate all missing angles.
C
D
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60o
E
25o
A
B
14
Angles in a Semi-Circle
General
Finish Maths in Action Worksheet 2 Then Page
114 Ex3 and Ex4
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15
Starter Questions
General
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16
Angles in a Semi-Circle
General
Pythagoras Theorem
Aim of Todays Lesson
How we can use Pythagoras Theorem to calculate
length within a circle
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17
Angles in a Semi-Circle
General
Pythagoras Theorem
We have been interested in right angled triangles
within a semi-circle. Since they are right angled
we can use Pythagoras Theorem to calculate
lengths.
Example 1 Calculate the value of d
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18
Angles in a Semi-Circle
General
Pythagoras Theorem
Example 2 Calculate the length of XY
Y
12cm
cm
X
Z
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13 cm
19
Angles in a Semi-Circle
General
Pythagoras Theorem
MIA Page 116 Ex5
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20
Starter Questions
General
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21
Angles in a Semi-Circle
General
SOHCAHTOA
Aim of Todays Lesson
How we can use SOHCAHTOA to calculate length and
angles within a circle
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22
Angles in a Semi-Circle
General
SOHCAHTOA
We have been interested in right angled triangles
within a semi-circle. Since they are right angled
we can use SOHCAHTOA to calculate lengths and
angles.
Example 1 Calculate the value of angle xo
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23
Angles in a Semi-Circle
General
SOHCAHTOA
Example 2 Calculate the length of AB
B
67.4o
A
C
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13 cm
24
Angles in a Semi-Circle
General
SOHCAHTOA
Maths In Action Page 117 Ex6
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25
Starter Questions
General
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26
Tangent to the circle
General
Aim of Todays Lesson
To understand what a tangent line is and its
special property with radius at the point of
contact.
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27
Tangent to the circle
General
A tangent line is a line that touches a circle
at only one point.
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28
Tangent to the circle
General
A tangent line is a line that touches a circle
at only one point.
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29
Tangent to the circle
General
The radius of the circle that touches the tangent
line is called the point of contact radius.
Special Property The point of contact radius is
always perpendicular (right-angled) to the
tangent line.
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30
Tangent to the circle
General
Q. Find the length of the tangent line between
A and B.
B
10
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By Pythagoras Theorem we have
C
A
8
31
Tangent to the circle
General
Begin Maths in Action Book Ex7A Ex7B page 185
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