Starter Draw a right angled triangle, where the two shorter sides are 7cm and 13cm, and measure the hypotenuse 7cm 13cm ?
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: Pythagoras’ theorem Using sides and angles: Trigonometry Right-angled triangles
Using just the sides: Pythagoras’ theorem The theorem Using sides and angles: Trigonometry Right-angled triangles
Using just the sides: Pythagoras’ theorem The theorem Finding the hypotenuse Using sides and angles: Trigonometry Right-angled triangles
Using just the sides: Pythagoras’ theorem The theorem Finding the hypotenuse Finding a shorter side Using sides and angles: Trigonometry Right-angled triangles
Using just the sides: Pythagoras’ theorem The theorem Finding the hypotenuse Finding a shorter side Pythagoras in 3-D Using sides and angles: Trigonometry Right-angled triangles
Using just the sides: Pythagoras’ theorem The theorem Finding the hypotenuse Finding a shorter side Pythagoras in 3-D Pythagoras and the circle Using sides and angles: Trigonometry Right-angled triangles
Using just the sides: Pythagoras’ theorem The theorem Finding the hypotenuse Finding a shorter side Pythagoras in 3-D Pythagoras and the circle Using sides and angles: Trigonometry Sine, cosine and tangent Right-angled triangles
Using just the sides: Pythagoras’ theorem The theorem Finding the hypotenuse Finding a shorter side Pythagoras in 3-D Pythagoras and the circle Using sides and angles: Trigonometry Sine, cosine and tangent Finding an angle Right-angled triangles
Using just the sides: Pythagoras’ theorem The theorem Finding the hypotenuse Finding a shorter side Pythagoras in 3-D Pythagoras and the circle Using sides and angles: Trigonometry Sine, cosine and tangent Finding an angle Finding a side Right-angled triangles
Using just the sides: Pythagoras’ theorem The theorem Finding the hypotenuse Finding a shorter side Pythagoras in 3-D Pythagoras and the circle Using sides and angles: Trigonometry Sine, cosine and tangent Finding an angle Finding a side Looking at the graphs Right-angled triangles
Using just the sides: Pythagoras’ theorem The theorem Finding the hypotenuse Finding a shorter side Pythagoras in 3-D Pythagoras and the circle Using sides and angles: Trigonometry Sine, cosine and tangent Finding an angle Finding a side Looking at the graphs Trigonometry beyond 90° Right-angled triangles
Using just the sides: Pythagoras’ theorem The theorem Finding the hypotenuse Finding a shorter side Pythagoras in 3-D Pythagoras and the circle Using sides and angles: Trigonometry Sine, cosine and tangent Finding an angle Finding a side Looking at the graphs Trigonometry beyond 90° Right-angled triangles
Pythagoras’ Theorem
Pythagoras’ Theorem
Pythagoras’ Theorem
Pythagoras’ Theorem
Pythagoras’ Theorem
Pythagoras’ Theorem
Pythagoras’ Theorem
Pythagoras’ Theorem
= 169 Pythagoras’ Theorem
a c b
a c b a2a2 b2b2 c2c2
a c b a2a2 b2b2 c2c2 a 2 + b 2 = c 2 Pythagoras’ Theorem
Finding the hypotenuse 4cm 3cm a 24cm 7cm b 13cm c
a 2 = cm 3cm a 24cm 7cm b 13cm c Finding the hypotenuse
a 2 = a 2 = cm 3cm a 24cm 7cm b 13cm c Finding the hypotenuse
a 2 = a 2 = a 2 = 25 4cm 3cm a 24cm 7cm b 13cm c Finding the hypotenuse
a 2 = a 2 = a 2 = 25 a = 5 cm 4cm 3cm a 24cm 7cm b 13cm c Finding the hypotenuse
a 2 = a 2 = a 2 = 25 a = 5 cm b 2 = cm 3cm a 24cm 7cm b 13cm c Finding the hypotenuse
a 2 = a 2 = a 2 = 25 a = 5 cm b 2 = b 2 = cm 3cm a 24cm 7cm b 13cm c Finding the hypotenuse
a 2 = a 2 = a 2 = 25 a = 5 cm b 2 = b 2 = b 2 = 625 4cm 3cm a 24cm 7cm b 13cm c Finding the hypotenuse
a 2 = a 2 = a 2 = 25 a = 5 cm b 2 = b 2 = b 2 = 625 b = 25 cm 4cm 3cm a 24cm 7cm b 13cm c Finding the hypotenuse
a 2 = a 2 = a 2 = 25 a = 5 cm b 2 = b 2 = b 2 = 625 b = 25 cm c 2 = cm 3cm a 24cm 7cm b 13cm c Finding the hypotenuse
a 2 = a 2 = a 2 = 25 a = 5 cm b 2 = b 2 = b 2 = 625 b = 25 cm c 2 = c 2 = cm 3cm a 24cm 7cm b 13cm c Finding the hypotenuse
a 2 = a 2 = a 2 = 25 a = 5 cm b 2 = b 2 = b 2 = 625 b = 25 cm c 2 = c 2 = c 2 = 218 4cm 3cm a 24cm 7cm b 13cm c Finding the hypotenuse
a 2 = a 2 = a 2 = 25 a = 5 cm b 2 = b 2 = b 2 = 625 b = 25 cm c 2 = c 2 = c 2 = 218 c = 14.8 cm (to 1 d.p.) 4cm 3cm a 24cm 7cm b 13cm c Finding the hypotenuse
a 2 = a 2 = a 2 = 25 a = 5 cm b 2 = b 2 = b 2 = 625 b = 25 cm c 2 = c 2 = c 2 = 218 c = 14.8 cm (to 1 d.p.) Exercise 15B Page 278 Questions 1-4 Finding the hypotenuse
Finding a shorter side a 3cm 5cm 9cm b 11cm
5 2 = a a 3cm 5cm 9cm b 11cm Finding a shorter side
5 2 = a = a a 3cm 5cm 9cm b 11cm Finding a shorter side
5 2 = a = a a 2 = 16 a 3cm 5cm 9cm b 11cm Finding a shorter side
5 2 = a = a a 2 = 16 a = 4 cm a 3cm 5cm 9cm b 11cm Finding a shorter side
5 2 = a = a a 2 = 16 a = 4 cm 11 2 = b a 3cm 5cm 9cm b 11cm Finding a shorter side
5 2 = a = a a 2 = 16 a = 4 cm 11 2 = b = b a 3cm 5cm 9cm b 11cm Finding a shorter side
5 2 = a = a a 2 = 16 a = 4 cm 11 2 = b = b b 2 = 40 a 3cm 5cm 9cm b 11cm Finding a shorter side
5 2 = a = a a 2 = 16 a = 4 cm 11 2 = b = b b 2 = 40 b = 6.3 cm (to 1 d.p.) a 3cm 5cm 9cm b 11cm Finding a shorter side
5 2 = a = a a 2 = 16 a = 4 cm 11 2 = b = b b 2 = 40 b = 6.3 cm (to 1 d.p.) Exercise 15C Page 280 Questions 1-6 Finding a shorter side
Pythagoras’ theorem applied twice a 6cm 5cm b 14cm
a 2 = a 2 = a 2 = 61 a = 7.8 cm a 6cm 5cm b 14cm Pythagoras’ theorem applied twice
a 2 = a 2 = a 2 = 61 a = 7.8 cm 14 2 = b = b b 2 = 135 b = 11.6 cm (to 1 d.p.) a 6cm 5cm b 14cm Pythagoras’ theorem applied twice
a 2 = a 2 = a 2 = 61 a = 7.8 cm 14 2 = b = b b 2 = 135 b = 11.6 cm (to 1 d.p.) a 6cm 5cm b 14cm Pythagoras’ theorem applied twice
a 2 = a 2 = a 2 = 61 a = 7.8 cm 14 2 = b = b b 2 = 135 b = 11.6 cm (to 1 d.p.) Exercise 15D Page 281 Question 1 Pythagoras’ theorem applied twice
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) ? The first number: How far apart are 2 & 11? 9 9 The distance between two points
What is the distance between (2, 3) and (11, -2) ? The first number: How far apart are 2 & 11? 9 The second number: How far apart are 3 & -2? 9 The distance between two points
What is the distance between (2, 3) and (11, -2) ? The first number: How far apart are 2 & 11? 9 The second number: How far apart are 3 & -2? The distance between two points
x 2 = x 2 = x 2 = 106 x = 10.3 units What is the distance between (2, 3) and (11, -2) ? The first number: How far apart are 2 & 11? 9 The second number: How far apart are 3 & -2? The distance between two points
x 2 = x 2 = x 2 = 106 x = 10.3 units What is the distance between (2, 3) and (11, -2) ? The first number: How far apart are 2 & 11? 9 The second number: How far apart are 3 & -2? 5 Exercise 15E Page 282 Questions 1-3 The distance between two points
Pythagoras’ theorem in 3- D
4 5 8
4 5 8 H E F G D C B A
4 5 8 H E F G D C B A
4 5 8 H E F G D C B A 5 4 E FG H EG 2 = EG 2 = EG 2 = 41 EG = 6.4 cm
Pythagoras’ theorem in 3- D H E F G D C B A EG 2 = EG 2 = EG 2 = 41 EG = 6.4 cm A E G AG 2 = AG 2 = AG 2 = 105 AG = 10.2 cm
Pythagoras’ theorem in 3- D H E F G D C B A EG 2 = EG 2 = EG 2 = 41 EG = 6.4 cm A E G AG 2 = AG 2 = AG 2 = 105 AG = 10.2 cm Note: = = 105
The equation of a circle
x 2 + y 2 = r 2