Circles

1
Circles
National 4
Revision of Angle Properties
Isosceles Triangles in Circles
Angles in a semi-circle
Pythagoras Theorem
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SOHCAHTOA
Tangent line on a circle
2
Starter Questions
National 4
B
6
co
C
A
8
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Revision Angle Properties
National 4
Learning Intention
Success Criteria

1. To know the basic properties for angles.

1. To review angle properties.

1. Solve problems using properties.

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4
Revision Angle Properties
National 4
65o
Two angles making a straight line add to 180o
DEMO
145o
Angles round a point Add up to 360o
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90o
146o
146o
3 angles in a triangle ALWAYS add up to 180o.
angles opposite each other at a cross are equal.
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RevisionAngle Properties
National 4
ALL angles in an equilateral triangle are 60o
Two angles in a isosceles Are equal
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h
is corresponding to d and must be 115o
b
is opposite to d and must be 115o
c
is must be 65o (straight line)
e
is alternate to c and must also be 65o
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Revision Angle Properties
National 4
Now try Ex 1 Ch12 (page 135) Copy out shapes
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7
Starter Questions
National 4
B
3
C
A
4
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Revision Angle Properties
National 4
Learning Intention
Success Criteria

1. To identify isosceles triangles with circles.

1. To investigate isosceles triangles with circles.

1. Solve problems using angle properties.

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Isosceles triangles in Circles
When two radii are drawn to the ends of a chord,
An isosceles triangle is formed.
DEMO
A
B
xo
xo
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C
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Isosceles triangles in Circles
Special Properties of Isosceles Triangles
Two equal lengths
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Two equal angles
Angles in any triangle sum to 180o
11
Isosceles triangles in Circles
Q. Find the angle xo.
B
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C
xo
Since the triangle is isosceles we have
A
280o
12
Revision Angle Properties
National 4
Now try Ex 1 Ch12 (page 137) Copy out shapes
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13
Starter Questions
National 4
B
bo
5
C
A
12
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27o
14
Angles in a Semi Circle
National 4
Learning Intention
Success Criteria

1. To know the special angle in a semi-circle.

1. To investigate the special angle in a semi-circle.

1. Solve problems using the special property.

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15
Angles in a Semi Circle
Tool-kit required
1. Protractor
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2. Pencil
3. Ruler
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Angles in a Semi Circle
1. Using your pencil trace round the protractor
so that you have semi-circle.
2. Mark the centre of the semi-circle.
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Angles in a Semi Circle
x
x
x
x

• Mark three points
• Outside the circle

x
x
x
x
x
2. On the circumference
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3. Inside the circle
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Angles in a Semi Circle
For each of the points Form a triangle by
drawing a line from each end of the diameter to
the point. Measure the angle at the various
points.
x
x
x
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Angles in a Semi Circle
DEMO
x
x
x
90o
gt 90o
lt 90o
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20
Angles in a Semi-Circle
National 4
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
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Angles in a Semi-Circle
National 4
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
22
Angles in a Semi-Circle
National 4
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
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Angles in a Semi-Circle
National 4
Now try Ex 2 Ch12 (page 138) Sketch shapes
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Starter Questions
National 4
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Angles in a Semi Circle
Pythagoras Theorem
National 4
Learning Intention
Success Criteria

1. To know when to use Pythagoras Theorem in a
   circle.

1. To explain how we can use
Pythagoras Theorem to calculate length
within a circle.

1. Solve problems using Pythagoras Theorem.

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Angles in a Semi-Circle
Pythagoras Theorem
National 4
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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5cm
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Angles in a Semi-Circle
Pythagoras Theorem
National 4
Example 2 Calculate the length of XY
Y
8cm
6cm
cm
X
Z
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10 cm
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Angles in a Semi-Circle
Pythagoras Theorem
National 4
Now try Ex 3 Ch12 (page 140) Q1 to Q9 Sketch
shapes
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Starter Questions
SOHCAHTOA
National 4
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Angles in a Semi Circle
SOHCAHTOA
National 4
Learning Intention
Success Criteria

• To know when to use
• Trigonometry (SOHCAHTOA)
• to calculate length and angles
• within a circle.

1. To explain how we can use
Trigonometry (SOHCAHTOA) to calculate
length and angles within a circle.
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1. Solve problems using Trigonometry (SOHCAHTOA) .

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Angles in a Semi-Circle
SOHCAHTOA
National 4
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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36.9o
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Angles in a Semi-Circle
SOHCAHTOA
National 4
Example 2 Calculate the length of AB
B
5cm
67.4o
A
C
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13 cm
33
Angles in a Semi-Circle
SOHCAHTOA
Now try Ex 3 Ch12 (page 140) Q10 onwards Sketch
shapes
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34
Starter Questions
If a 7 b 4 and c 10 Write down as many
equations as you can
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e.g. a b c 21
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Angles in a Semi Circle
Tangent Line
Learning Intention
Success Criteria

1. To understand what a tangent line is.

• To explain how what a tangent
• line is and its special property with the
  radius at the point of contact.

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1. Solve problems using the tangent property.

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Angles in a Semi-Circle
Tangent Line
A tangent line is a line that touches a circle
at only one point.
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37
Angles in a Semi-Circle
Tangent Line
The radius of the circle that touches the tangent
line is called the point of contact radius.
DEMO
Special Property The point of contact radius is
always perpendicular (right-angled) to the
tangent line.
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Angles in a Semi-Circle
Tangent Line
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
39
Angles in a Semi-Circle
Tangent Line
Now try Work Sheet Sketch shapes
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