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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