Isosceles Triangles in Circles
1
AREA OF PART OF A CIRCLE
2
Isosceles Triangles in Circles
Right angle in a Semi-Circle
The Circle
Tangent Line to a Circle
Diameter Symmetry in a Circle
Circumference of a Circle
Length of an ARC of a Circle
Area of a Circle
Area of a SECTOR of a Circle
Summary of Circle Chapter
3
To identify isosceles triangles
within a circle.
Aim of Today’s Lesson
Isosceles triangles
Isosceles triangles
in Circles
in Circles
4
When two radii are drawn to the ends of a chord,
When two radii are drawn to the ends of a chord,
An isosceles triangle is formed.
An isosceles triangle is formed.
C
A
B
Isosceles triangles
Isosceles triangles
in Circles
in Circles
x
o
x
o
5
Special Properties of Isosceles Triangles
Two equal lengths
Two equal angles
Angles in any triangle sum to 180
o
Isosceles triangles
Isosceles triangles
in Circles
in Circles
6
Q.
Q.
Find the angle x
Find the angle x
o
o
.
.
A
B
C
Since the triangle is isosceles
we have
x
o
280
o
Isosceles triangles
Isosceles triangles
in Circles
in Circles
7
Isosceles triangles
Isosceles triangles
in Circles
in Circles
8
Semi-circle angle
To find the angle in a semi-circle
made by a triangle with hypotenuse
equal to the diameter and the two smaller
lengths meeting at the circumference.
Aim of Today’s Lesson
9
Semi-circle angle
Semi-circle angle
Tool-kit required
Tool-kit required
1.
1.
Protractor
Protractor
2.
2.
Pencil
Pencil
3.
3.
Ruler
Ruler
10
1.
1.
Using your pencil trace round
Using your pencil trace round
the protractor so that you have
the protractor so that you have
semi-circle.
semi-circle.
2.
2.
Mark the centre of
Mark the centre of
the semi-circle.
the semi-circle.
Semi-circle angle
Semi-circle angle
11
Mark three points
Mark three points
1.
Outside the circle
Outside the circle
x
x
x
x
x
x
x
x
x
Semi-circle angle
Semi-circle angle
2. On the circumference
2. On the circumference
3. Inside the circle
3. Inside the circle
12
For each of the points
For each of the points
Form a triangle by drawing a
Form a triangle by drawing a
line from each end of the
line from each end of the
diameter to the point.
diameter to the point.
Measure the angle at the
Measure the angle at the
various points.
various points.
x
x
x
Semi-circle angle
Semi-circle angle
13
x
Semi-circle angle
Semi-circle angle
x
x
< 90
o
> 90
o
= 90
= 90
o
o
14
To understand what a tangent line is
and its special property with the
radius at the point of contact.
Aim of Today’s Lesson
Tangent line
Tangent line
15
Tangent line
Tangent line
A
A
tangent line
tangent line
is a line that
is a line that
touches a circle at
touches a circle at
only one point.
only one point.
16
Tangent line
Tangent line
The radius of the circle that touches the tangent
The radius of the circle that touches the tangent
line is called
line is called
the point of contact radius.
the point of contact radius.
Special Property
Special Property
The point of contact radius
The point of contact radius
is always perpendicular
is always perpendicular
(right-angled)
(right-angled)
to the tangent line.
to the tangent line.
17
Tangent line
Tangent line
Q.
Q.
Find the length of the tangent line between
Find the length of the tangent line between
A and B.
A and B.
A
B
8
10
C
By Pythagoras Theorem we have
18
Tangent line
Tangent line
19
To understand
some special properties
when a diameter
bisects
a chord.
Aim of Today’s Lesson
Diameter symmetry
Diameter symmetry
20
Diameter symmetry
Diameter symmetry
1.
A line drawn through the centre of a circle
through the midpoint a chord will
ALWAYS
cut
the chord at right-angles
2.
A line drawn through the centre of a circle
at right-angles to a chord will
ALWAYS
bisect that chord.
3.
A line bisecting a chord at right angles
will
ALWAYS
pass through the centre of a circle.
O
21
Q.
Q.
Find the length of the chord A and B.
Find the length of the chord A and B.
A
B
6
O
Solution
By Pythagoras Theorem we have
Diameter symmetry
Diameter symmetry
4
Since yellow line bisect AB and passes
through centre O, triangle is right-angle.
Radius of the circle is 4 + 6 = 10.
10
Since AB is bisected
The length of AB is
22
Diameter symmetry
Diameter symmetry
23
Circumference
of a circle
Aim of Today’s Lesson
To be able to use the formula
for calculating
the circumference of a circle
24
Main parts of the circle
O
Circumference
of a circle
25
Q.
Find the circumference of the circle ?
4cm
Circumference
of a circle
26
Q.
The circumference of the circle is 60cm ?
Find the length of the diameter and radius.
Circumference
of a circle
27
Circumference
of a circle
28
length of the
arc of a circle
Aim of Today’s Lesson
To find and be able to use the formula
for calculating the length of an arc.
29
Q.
What is an arc ?
A
B
Answer
Answer
An arc is a fraction
of the circumference.
Arc length of a circle
30
Q.
Find the circumference of the circle ?
10cm
Arc length of a circle
31
Arc length of a circle
A
r
c
l
e
n
g
t
h
π
D
A
r
c
a
n
g
l
e
3
6
0
o
=
Q.
Find the length of the minor arc XY below ?
6 cm
45
o
x
y
3
6
0
o
c
o
n
n
e
c
t
i
o
n
32
Arc length of a circle
Q.
Find the length of the minor arc AB below ?
9 cm
60
o
A
B
c
o
n
n
e
c
t
i
o
n
33
Arc length of a circle
Q.
Find the length of the major arc PQ below ?
10 m
100
o
P
Q
c
o
n
n
e
c
t
i
o
n
2
6
0
o
34
length of the
arc of a circle
35
The Area of a circle
Aim of Today’s Lesson
To come up with and be able to use
the formula for calculating
the area of a circle
36
The Area of a circle
37
The Area of a circle
38
But the area inside this rectangle is also the area of the circle
The Area of a circle
39
Q.
Find the area of the circle ?
4cm
The Area of a circle
40
The Area
The Area
of a circle
of a circle
41
Q.
The diameter of the circle is 60cm.
Find area of the circle?
The Area of a circle
42
The Area
The Area
of a circle
of a circle
43
Q.
The area of a circle is 12.64 cm
2
.
Find its radius?
The Area of a circle
44
Sector area of a circle
Aim of Today’s Lesson
To find and be able to use the formula
for calculating the sector of an circle.
45
A
B
Area of Sector in a circle
46
Q.
Find the area of the circle ?
10cm
Area of Sector in a circle
47
A
r
e
a
S
e
c
t
o
r
π
r
2
S
e
c
t
o
r
a
n
g
l
e
3
6
0
o
=
Find the area of the minor sector XY below ?
6 cm
45
o
x
y
3
6
0
o
c
o
n
n
e
c
t
i
o
n
Area of Sector in a circle
48
Q.
Find the area of the minor sector AB below ?
9 cm
A
B
c
o
n
n
e
c
t
i
o
n
Area of Sector in a circle
60
o
49
Area of Sector in a circle
Q.
Find the area of the major sector PQ below ?
10 m
100
o
P
Q
c
o
n
n
e
c
t
i
o
n
2
6
0
o
50
Sector area of a circle
Sector area of a circle
51
Summary of Circle Topic
Summary of Circle Topic
line that bisects a chord
1.
Splits the chord into 2
equal halves.
2.
Makes right-angle with
the chord.
3.
Passes through centre
of the circle
Semi-circle angle is
always 90
o
Tangent touches
circle at one point
and make angle 90
o
with point of contact
radius
Exploring the properties and characteristics of isosceles triangles within circles, focusing on identifying, understanding angles, and special properties within these geometric shapes. Key concepts covered include equal lengths, equal angles, and angle calculations within isosceles triangles forming in circles.
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Presentation Transcript
The Circle Isosceles Triangles in Circles Right angle in a Semi-Circle Tangent Line to a Circle Diameter Symmetry in a Circle Circumference of a Circle Length of an ARC of a Circle Area of a Circle Area of a SECTOR of a Circle Summary of Circle Chapter 2
Isosceles triangles in Circles Aim of Today s Lesson To identify isosceles triangles within a circle. 3
Isosceles triangles in Circles When two radii are drawn to the ends of a chord, An isosceles triangle is formed. A B xo xo C 4
Isosceles triangles in Circles Special Properties of Isosceles Triangles Two equal lengths Two equal angles Angles in any triangle sum to 180o 5
Isosceles triangles in Circles o. Q. Find the angle x Solution Angle at C is equal to: B 360 280 80 o o o = o C x Since the triangle is isosceles we have 2 x + A 280o o o o = 80 180 o o x = 2 100 o o x = 50 6
Isosceles triangles in Circles 7
Semi-circle angle Aim of Today s Lesson To find the angle in a semi-circle made by a triangle with hypotenuse equal to the diameter and the two smaller lengths meeting at the circumference. 8
Semi-circle angle Tool-kit required 1. Protractor 2. Pencil 3. Ruler 9
Semi-circle angle 1. Using your pencil trace round the protractor so that you have semi-circle. 2. Mark the centre of the semi-circle. You should have something like this. 10
Semi-circle angle x x x x Mark three points x 1. Outside the circle 2. On the circumference 3. Inside the circle x x x x 11
Semi-circle angle x For each of the points x Form a triangle by drawing a line from each end of the diameter to the point. Measure the angle at the various points. Log your results in a table. Circumference Outside x Inside 12
Semi-circle angle x x x Outside < 90 Inside > 90 Circumference = 90 o o o 13
Tangent line Aim of Today s Lesson To understand what a tangent line is and its special property with the radius at the point of contact. 14
Tangent line A tangent line is a line that touches a circle at only one point. Which of the lines are tangent to the circle? 15
Tangent line 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. 16
Tangent line Q. Solution Right-angled at A since AC is the radius at the point of contact with the Tangent. By Pythagoras Theorem we have + a b + 8 a a a Find the length of the tangent line between A and B. B 10 C A 8 c 10 10 100 36 2 2 2 = = = = 2 2 2 8 2 2 2 64 6 36 2 = a = = 17
Tangent line 18
Diameter symmetry Aim of Today s Lesson To understand some special properties when a diameter bisects a chord. 19
Diameter symmetry C 1. A line drawn through the centre of a circle through the midpoint a chord will ALWAYS cut the chord at right-angles 2. A line drawn through the centre of a circle at right-angles to a chord will ALWAYS bisect that chord. O A B 3. A line bisecting a chord at right angles will ALWAYS pass through the centre of a circle. D 20
Diameter symmetry Q. Find the length of the chord A and B. Solution B Radius of the circle is 4 + 6 = 10. Since yellow line bisect AB and passes through centre O, triangle is right-angle. 10 O By Pythagoras Theorem we have + = a b c + = 6 10 a 4 6 2 2 2 Since AB is bisected The length of AB is 2 2 2 A a a 10 100 36 64 6 2 2 2 = = = 64 2 = length 2 8 16 = = a 8 = AB 21
Circumference of a circle Aim of Today s Lesson To be able to use the formula for calculating the circumference of a circle 23
Circumference of a circle Main parts of the circle radius O Diameter 2 D = r Circumference = C D 24
Circumference of a circle Q. Find the circumference of the circle ? Solution = = = 4cm C C C D 8 25.12 cm 25
Circumference of a circle Q. The circumference of the circle is 60cm ? Find the length of the diameter and radius. Solution = = 60 =60 D C D D 2 r = D 19 2r 19 2 9.5 = cm r = = r cm D 19 cm 26
Circumference of a circle 27
length of the arc of a circle Aim of Today s Lesson To find and be able to use the formula for calculating the length of an arc. 28
Arc length of a circle Q. What is an arc ? Answer A An arc is a fraction of the circumference. minor arc B major arc 29
Arc length of a circle Find the circumference of the circle ? Q. Solution = = = 10cm C C C D 20 62.8 cm 30
Arc length of a circle Q. Find the length of the minor arc XY below ? x connection Arc length D Arc angle 360o = y 6 cm o 45 45 360 o arc length ( 12) = o o 360 arc length 4.71 cm = 31
Arc length of a circle Q. Find the length of the minor arc AB below ? Arc length D Arc angle 360o = connection A 9 cm 60 360 o o 60 arc length ( 18) = o B arc length 9.42 cm = 32
Arc length of a circle Q. Find the length of the major arc PQ below ? Arc length D Arc angle 360o = connection P 10 m 260 360 o o o 260 100 arc length ( 20) = o Q arc length 45.38 cm = 33
length of the arc of a circle 34
The Area of a circle Aim of Today s Lesson To come up with and be able to use the formula for calculating the area of a circle 35
The Area of a circle If we break the circle into equal sectors And lay them out side by side We get very close to a rectangle. 1 8 2 7 3 6 5 8 6 4 4 2 3 7 1 5 36
The Area of a circle If we cut the sectors Thinner and thinner then we get closer and closer to a rectangle. Hence we can represent the area of a circle by a rectangle. 8 6 4 23 7 1 5 thinner and thinner sectors r r 37
The Area of a circle r r Area of a rectangle Area of a rectangle l b = r r r 2 = = But the area inside this rectangle is also the area of the circle Area of a circle r 2 = 38
The Area of a circle Q. Find the area of the circle ? Solution A = = A A = 4cm r 2 50.26 4 2 cm 2 39
The Area of a circle Q. The diameter of the circle is 60cm. Find area of the circle? Solution A = D = r 2 60 2 2827.43 r 30 cm = = 2 A 30 2 = A cm 2 = 41
The Area of a circle 2. Q. The area of a circle is 12.64 cm Find its radius? Solution A = = r 2 12.64 r 2 12.64 r 4 cm 2 = = 4 r 2 cm = = 43
Sector area of a circle Aim of Today s Lesson To find and be able to use the formula for calculating the sector of an circle. 44
Area of Sector in a circle A minor sector B major sector 45
Area of Sector in a circle Find the area of the circle ? Q. Solution A = = A A = 10cm r 2 314 10 cm 2 2 46
Area of Sector in a circle Find the area of the minor sector XY below ? x connection Area Sector r2 Sector angle 360o = y 6 cm o 45 45 360 o Area of Sector ( 6 ) 2 = o o 360 Area Sector 14.14 cm 2 = 47
Area of Sector in a circle Q. Find the area of the minor sector AB below ? connection Area Sector r2 Sector angle 360o A = 9 cm 60 360 o o 60 Area Sector ( 9 ) 2 = o B Area Sector 42.41 cm 2 = 48
Area of Sector in a circle Q. Find the area of the major sector PQ below ? connection Sector Area r2 Sector angle 360o = P 10 m 260 360 o o o 260 100 Sector Area ( 10 ) 2 = o Q Area Sector 226.89 cm 2 = 49
