Download Chp 15 & 16 - gcse-maths

Starter
Draw a right angled triangle, where
the two shorter sides are 7cm and
13cm, and measure the hypotenuse
7cm
?
13cm
Right-angled triangles
Right-angled triangles
Using just the sides:
Using sides and angles:
Right-angled triangles
Using just the sides:
Pythagoras’ theorem
Using sides and angles:
Right-angled triangles
Using just the sides:
Using sides and angles:
Pythagoras’ theorem
Trigonometry
Right-angled triangles
Using just the sides:
Using sides and angles:
Pythagoras’ theorem
Trigonometry
• The theorem
Right-angled triangles
Using just the sides:
Using sides and angles:
Pythagoras’ theorem
Trigonometry
• The theorem
• Finding the hypotenuse
Right-angled triangles
Using just the sides:
Using sides and angles:
Pythagoras’ theorem
Trigonometry
• The theorem
• Finding the hypotenuse
• Finding a shorter side
Right-angled triangles
Using just the sides:
Using sides and angles:
Pythagoras’ theorem
Trigonometry
• The theorem
• Finding the hypotenuse
• Finding a shorter side
• Pythagoras in 3-D
Right-angled triangles
Using just the sides:
Using sides and angles:
Pythagoras’ theorem
Trigonometry
• The theorem
• Finding the hypotenuse
• Finding a shorter side
• Pythagoras in 3-D
• Pythagoras and the circle
Right-angled triangles
Using just the sides:
Using sides and angles:
Pythagoras’ theorem
Trigonometry
• The theorem
• Finding the hypotenuse
• Finding a shorter side
• Pythagoras in 3-D
• Pythagoras and the circle
• Sine, cosine and tangent
Right-angled triangles
Using just the sides:
Using sides and angles:
Pythagoras’ theorem
Trigonometry
• The theorem
• Sine, cosine and tangent
• Finding the hypotenuse
• Finding an angle
• Finding a shorter side
• Pythagoras in 3-D
• Pythagoras and the circle
Right-angled triangles
Using just the sides:
Using sides and angles:
Pythagoras’ theorem
Trigonometry
• The theorem
• Sine, cosine and tangent
• Finding the hypotenuse
• Finding an angle
• Finding a shorter side
• Finding a side
• Pythagoras in 3-D
• Pythagoras and the circle
Right-angled triangles
Using just the sides:
Using sides and angles:
Pythagoras’ theorem
Trigonometry
• The theorem
• Sine, cosine and tangent
• Finding the hypotenuse
• Finding an angle
• Finding a shorter side
• Finding a side
• Pythagoras in 3-D
• Looking at the graphs
• Pythagoras and the circle
Right-angled triangles
Using just the sides:
Using sides and angles:
Pythagoras’ theorem
Trigonometry
• The theorem
• Sine, cosine and tangent
• Finding the hypotenuse
• Finding an angle
• Finding a shorter side
• Finding a side
• Pythagoras in 3-D
• Looking at the graphs
• Pythagoras and the circle
• Trigonometry beyond 90°
Right-angled triangles
Using just the sides:
Using sides and angles:
Pythagoras’ theorem
Trigonometry
• The theorem
• Sine, cosine and tangent
• Finding the hypotenuse
• Finding an angle
• Finding a shorter side
• Finding a side
• Pythagoras in 3-D
• Looking at the graphs
• Pythagoras and the circle
• Trigonometry beyond 90°
Pythagoras’ Theorem
5
13
12
Pythagoras’ Theorem
5
13
12
Pythagoras’ Theorem
5
13
12
Pythagoras’ Theorem
5
13
12
Pythagoras’ Theorem
5
5
13
13
12
12
Pythagoras’ Theorem
5
25
5
13
13
12
12
Pythagoras’ Theorem
5
25
5
13
13
12
144
12
Pythagoras’ Theorem
169
5
25
5
13
13
12
144
12
Pythagoras’ Theorem
169
5
25
5
13
13
12
25 + 144 = 169
144
12
Pythagoras’ Theorem
a
c
b
Pythagoras’ Theorem
c2
a2 a
c
b
b2
Pythagoras’ Theorem
c2
a2 a
c
b
a2
+
b2
=
c2
b2
Finding the hypotenuse
24cm
3cm
a
b
4cm
7cm
7cm
13cm
c
Finding the hypotenuse
24cm
3cm
a
b
4cm
a2 = 3 2 + 4 2
7cm
7cm
13cm
c
Finding the hypotenuse
24cm
3cm
a
b
4cm
a2 = 3 2 + 4 2
a2 = 9 + 16
7cm
7cm
13cm
c
Finding the hypotenuse
24cm
3cm
a
b
4cm
a2 = 3 2 + 4 2
a2 = 9 + 16
a2 = 25
7cm
7cm
13cm
c
Finding the hypotenuse
24cm
3cm
a
b
4cm
a2 = 3 2 + 4 2
a2 = 9 + 16
a2 = 25
a = 5 cm
7cm
7cm
13cm
c
Finding the hypotenuse
24cm
3cm
a
b
4cm
a2 = 3 2 + 4 2
a2 = 9 + 16
a2 = 25
a = 5 cm
7cm
7cm
13cm
b2 = 242 + 72
c
Finding the hypotenuse
24cm
3cm
a
b
4cm
13cm
a2 = 3 2 + 4 2
b2 = 242 + 72
a2 = 9 + 16
b2 = 576 + 49
a2 = 25
a = 5 cm
7cm
7cm
c
Finding the hypotenuse
24cm
3cm
a
b
4cm
13cm
a2 = 3 2 + 4 2
b2 = 242 + 72
a2 = 9 + 16
b2 = 576 + 49
a2 = 25
b2 = 625
a = 5 cm
7cm
7cm
c
Finding the hypotenuse
24cm
3cm
a
b
4cm
7cm
7cm
13cm
a2 = 3 2 + 4 2
b2 = 242 + 72
a2 = 9 + 16
b2 = 576 + 49
a2 = 25
b2 = 625
a = 5 cm
b = 25 cm
c
Finding the hypotenuse
24cm
3cm
a
b
4cm
7cm
7cm
c
13cm
a2 = 3 2 + 4 2
b2 = 242 + 72
a2 = 9 + 16
b2 = 576 + 49
a2 = 25
b2 = 625
a = 5 cm
b = 25 cm
c2 = 72 + 132
Finding the hypotenuse
24cm
3cm
a
b
4cm
7cm
7cm
c
13cm
a2 = 3 2 + 4 2
b2 = 242 + 72
c2 = 72 + 132
a2 = 9 + 16
b2 = 576 + 49
c2 = 49 + 169
a2 = 25
b2 = 625
a = 5 cm
b = 25 cm
Finding the hypotenuse
24cm
3cm
a
b
4cm
7cm
7cm
c
13cm
a2 = 3 2 + 4 2
b2 = 242 + 72
c2 = 72 + 132
a2 = 9 + 16
b2 = 576 + 49
c2 = 49 + 169
a2 = 25
b2 = 625
c2 = 218
a = 5 cm
b = 25 cm
Finding the hypotenuse
24cm
3cm
a
b
4cm
7cm
7cm
c
13cm
a2 = 3 2 + 4 2
b2 = 242 + 72
c2 = 72 + 132
a2 = 9 + 16
b2 = 576 + 49
c2 = 49 + 169
a2 = 25
b2 = 625
c2 = 218
a = 5 cm
b = 25 cm
c = 14.8 cm (to 1 d.p.)
Finding the hypotenuse
Exercise 15B Page 278 Questions 1-4
a2 = 3 2 + 4 2
b2 = 242 + 72
c2 = 72 + 132
a2 = 9 + 16
b2 = 576 + 49
c2 = 49 + 169
a2 = 25
b2 = 625
c2 = 218
a = 5 cm
b = 25 cm
c = 14.8 cm (to 1 d.p.)
Finding a shorter side
9cm
5cm
3cm
11cm
a
b
Finding a shorter side
9cm
5cm
3cm
11cm
a
52 = a 2 + 3 2
b
Finding a shorter side
9cm
5cm
3cm
11cm
a
52 = a 2 + 3 2
25 = a2 + 9
b
Finding a shorter side
9cm
5cm
3cm
11cm
a
52 = a 2 + 3 2
25 = a2 + 9
a2 = 16
b
Finding a shorter side
9cm
5cm
3cm
11cm
a
52 = a 2 + 3 2
25 = a2 + 9
a2 = 16
a = 4 cm
b
Finding a shorter side
9cm
5cm
3cm
11cm
a
52 = a 2 + 3 2
25 = a2 + 9
a2 = 16
a = 4 cm
112 = b2 + 92
b
Finding a shorter side
9cm
5cm
3cm
11cm
a
52 = a 2 + 3 2
112 = b2 + 92
25 = a2 + 9
121 = b2 + 81
a2 = 16
a = 4 cm
b
Finding a shorter side
9cm
5cm
3cm
11cm
a
52 = a 2 + 3 2
112 = b2 + 92
25 = a2 + 9
121 = b2 + 81
a2 = 16
b2 = 40
a = 4 cm
b
Finding a shorter side
9cm
5cm
3cm
11cm
b
a
52 = a 2 + 3 2
112 = b2 + 92
25 = a2 + 9
121 = b2 + 81
a2 = 16
b2 = 40
a = 4 cm
b = 6.3 cm (to 1 d.p.)
Finding a shorter side
Exercise 15C Page 280 Questions 1-6
52 = a 2 + 3 2
112 = b2 + 92
25 = a2 + 9
121 = b2 + 81
a2 = 16
b2 = 40
a = 4 cm
b = 6.3 cm (to 1 d.p.)
Pythagoras’ theorem applied twice
5cm
14cm
a
6cm
b
Pythagoras’ theorem applied twice
5cm
14cm
a
6cm
b
a2 = 5 2 + 6 2
a2 = 25 + 36
a2 = 61
a = 7.8 cm
Pythagoras’ theorem applied twice
5cm
14cm
a
6cm
b
a2 = 5 2 + 6 2
142 = b2 + 7.82
a2 = 25 + 36
196 = b2 + 61
a2 = 61
b2 = 135
a = 7.8 cm
b = 11.6 cm (to 1 d.p.)
Pythagoras’ theorem applied twice
5cm
14cm
a
6cm
b
a2 = 5 2 + 6 2
142 = b2 + 7.82
a2 = 25 + 36
196 = b2 + 61
a2 = 61
b2 = 135
a = 7.8 cm
b = 11.6 cm (to 1 d.p.)
Pythagoras’ theorem applied twice
Exercise 15D Page 281 Question 1
a2 = 5 2 + 6 2
142 = b2 + 7.82
a2 = 25 + 36
196 = b2 + 61
a2 = 61
b2 = 135
a = 7.8 cm
b = 11.6 cm (to 1 d.p.)
The distance between two points
What is the distance between (2, 3) and (11, -2) ?
The distance between two points
What is the distance between (2, 3) and (11, -2) ?
The distance between two points
What is the distance between (2, 3) and (11, -2) ?
The distance between two points
What is the distance between (2, 3) and (11, -2) ?
The first number:
How far apart are 2 & 11?
The distance between two points
What is the distance between (2, 3) and (11, -2) ?
9
The first number:
How far apart are 2 & 11? 9
The distance between two points
What is the distance between (2, 3) and (11, -2) ?
9
The first number:
How far apart are 2 & 11? 9
The second number:
How far apart are 3 & -2?
The distance between two points
What is the distance between (2, 3) and (11, -2) ?
5
9
The first number:
How far apart are 2 & 11? 9
The second number:
How far apart are 3 & -2? 5
The distance between two points
What is the distance between (2, 3) and (11, -2) ?
5
9
The first number:
x2 = 9 2 + 5 2
How far apart are 2 & 11? 9
x2 = 81 + 25
The second number:
x2 = 106
How far apart are 3 & -2? 5
x = 10.3 units
The distance between two points
Exercise 15E Page 282 Questions 1-3
What is the distance between (2, 3) and (11, -2) ?
The first number:
x2 = 9 2 + 5 2
How far apart are 2 & 11? 9
x2 = 81 + 25
The second number:
x2 = 106
How far apart are 3 & -2? 5
x = 10.3 units
Pythagoras’ theorem in 3-D
Pythagoras’ theorem in 3-D
8
5
4
Pythagoras’ theorem in 3-D
B
C
A
D
8
E
F
5
4
H
G
Pythagoras’ theorem in 3-D
B
C
A
D
8
E
F
5
4
H
G
Pythagoras’ theorem in 3-D
B
C
A
F
G
D
8
E
4
F
5
EG2 = 52 + 42
EG2 = 25 + 16
EG2 = 41
EG = 6.4 cm
H
4
G
E
5
H
Pythagoras’ theorem in 3-D
B
C
A
D
8
E
8
F
5
H
A
4
G
E
EG2 = 52 + 42
AG2 = 82 + 6.42
EG2 = 25 + 16
AG2 = 64 + 41
EG2 = 41
AG2 = 105
EG = 6.4 cm
AG = 10.2 cm
6.4
G
Pythagoras’ theorem in 3-D
B
C
A
D
8
E
8
F
5
H
A
4
G
E
6.4
G
EG2 = 52 + 42
AG2 = 82 + 6.42
Note:
EG2 = 25 + 16
AG2 = 64 + 41
52 + 4 2 + 8 2
EG2 = 41
AG2 = 105
= 25 + 16 + 64
EG = 6.4 cm
AG = 10.2 cm
= 105
The equation of a circle
The equation of a circle
The equation of a circle
x2 + y2 = r2