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KS4 Mathematics

KS4 Mathematics. S2 Pythagoras’ Theorem. S2 Pythagoras’ Theorem. Contents. A. S2.2 Identifying right-angled triangles. A. S2.3 Pythagorean triples. A. S2.1 Introducing Pythagoras’ Theorem. S2.4 Finding unknown lengths. A. S2.5 Applying Pythagoras’ Theorem in 2-D. A.

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KS4 Mathematics

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  1. KS4 Mathematics S2 Pythagoras’ Theorem

  2. S2 Pythagoras’ Theorem Contents • A S2.2 Identifying right-angled triangles • A S2.3 Pythagorean triples • A S2.1 Introducing Pythagoras’ Theorem S2.4 Finding unknown lengths • A S2.5 Applying Pythagoras’ Theorem in 2-D • A S2.6 Applying Pythagoras’ Theorem in 3-D • A

  3. Right-angled triangles A right-angled triangle contains a right angle. The longest side opposite the right angle is called the hypotenuse.

  4. Identify the hypotenuse

  5. The history of Pythagoras’ Theorem Pythagoras’ Theorem concerns the relationship between the sides of a right- angled triangle. The theorem is named after the Greek mathematician and philosopher, Pythagoras of Samos. Although the Theorem is named after Pythagoras, the result was known to many ancient civilizations including the Babylonians, Egyptians and Chinese, at least 1000 years before Pythagoras was born.

  6. Pythagoras’ Theorem Pythagoras’ Theorem states that the square formed on the hypotenuse of a right-angled triangle … … has the same area as the sum of the areas of the squares formed on the other two sides.

  7. Showing Pythagoras’ Theorem

  8. Pythagoras’ Theorem If we label the length of the sides of a right-angled triangle a, b and c as follows, then the area of the largest square is c × c or c2. The areas of the smaller squares are a2 and b2. c2 c a2 a We can write Pythagoras’ Theorem as b b2 c2 =a2 + b2

  9. A proof of Pythagoras’ Theorem

  10. S2 Pythagoras’ Theorem Contents S2.1 Introducing Pythagoras’ Theorem • A • A S2.3 Pythagorean triples • A S2.2 Identifying right-angled triangles S2.4 Finding unknown lengths • A S2.5 Applying Pythagoras’ Theorem in 2-D • A S2.6 Applying Pythagoras’ Theorem in 3-D • A

  11. Pythagoras’ Theorem Pythagoras’ Theorem states that for a right-angled triangle with a hypotenuse of length c and the shorter sides of lengths a and b c a c2 =a2 + b2 b We can use Pythagoras’ Theorem • to check whether a triangle is right-angled given the lengths of all the sides, • to find the length of a missing side in a right-angled triangle given the lengths of the other two sides.

  12. Identifying right-angled triangles If we are given the lengths of the three sides of a triangle, we can use Pythagoras’ Theorem to find out whether or not the triangle contains a right angle. For example: A triangle has sides of length 4 cm, 7 cm and 9 cm. Is this a right-angled triangle? If the sum of the squares on the two shorter sides is equal to the square on the longest side, the triangle has a right angle. 42 + 72 = 16 + 49 = 65 65  92 No, this is not a right-angled triangle.

  13. Identifying right-angled triangles

  14. Identifying right-angled triangles If the sum of the squares on the two shortest sides of a triangle is greater than the sum of the square on the longest side, all the angles in the triangle must be acute. In other words, if a triangle with sides of length a, b and c, where c is the longest side has, c2 <a2 + b2 then all the angles in the triangle must be acute.

  15. Identifying right-angled triangles If the sum of the squares on the two shortest sides of a triangle is less than the sum of the square on the longest side, one of the angles in the triangle must be obtuse. In other words, if a triangle with sides of length a, b and c, where c is the longest side has, c2 >a2 + b2 then one of the angles in the triangle must be obtuse.

  16. Identifying right-angled triangles 10 cm 5 cm ? 8 cm A triangle has sides of length 5 cm, 8 cm and 10 cm. Is the angle opposite the longest side acute, obtuse or right-angled? 82 + 52 = 64 + 25 = 89 102 = 100 100 > 89 so the angle opposite the longest side is an obtuse angle.

  17. Obtuse, acute or right-angled triangle

  18. S2 Pythagoras’ Theorem Contents S2.1 Introducing Pythagoras’ Theorem • A S2.2 Identifying right-angled triangles • A • A S2.3 Pythagorean triples S2.4 Finding unknown lengths • A S2.5 Applying Pythagoras’ Theorem in 2-D • A S2.6 Applying Pythagoras’ Theorem in 3-D • A

  19. Pythagorean triples A triangle has sides of length 3 cm, 4 cm and 5 cm. Does this triangle have a right angle? Using Pythagoras’ Theorem, if the sum of the squares on the two shorter sides is equal to the square on the longest side, the triangle has a right angle. 32 + 42 = 9 + 16 = 25 = 52 Yes, the triangle has a right-angle. The numbers 3, 4 and 5 form a Pythagorean triple.

  20. Pythagorean triples Ancient Egyptians used the fact that a triangle with sides of length 3, 4 and 5 contained a right-angle to mark out field boundaries and for building.

  21. Pythagorean triples Write down every square number from 12 = 1 to 202 = 400. Use these numbers to find as many Pythagorean triples as you can. Write down any patterns that you notice. Three whole numbers a, b and c, where c is the largest, form a Pythagorean triple if, a2 + b2 = c2 3, 4, 5 is the simplest Pythagorean triple.

  22. Pythagorean triples How many of these did you find? 9 + 16 = 25 32 + 42 = 52 3, 4, 5 36 + 64 = 100 62 + 82 = 102 6, 8, 10 25 + 144 = 169 52 + 122 = 132 5, 12, 13 81 + 144 = 225 92 + 122 = 152 9,12, 15 64 + 225 = 289 82 + 152 = 172 8, 15, 17 144 + 256 = 400 122 + 162 = 202 12, 16, 20 The Pythagorean triples 3, 4, 5; 5, 12, 13 and 8, 15 17 are called primitive Pythagorean triples because they are not multiples of another Pythagorean triple.

  23. Similar right-angled triangles The following right-angled triangles are similar. 15 10 ? 6 9 8 12 ? Find the lengths of the missing sides. Check that Pythagoras’ Theorem holds for both triangles.

  24. Similar right-angled triangles The following right-angled triangles are similar. 15 10 ? 6 9 8 12 ? 62 + 82 = 36 + 64 92 + 122 = 81 + 144 = 100 = 225 = 102 = 152

  25. Similar right-angled triangles The following right-angled triangles are similar. 60 45 ? 32 ? 24 68 51 Find the lengths of the missing sides. Check that Pythagoras’ Theorem holds for both triangles.

  26. Similar right-angled triangles The following right-angled triangles are similar. 60 45 32 24 68 51 322 + 602 = 1024 + 3600 242 + 452 = 576 + 2025 = 4624 = 2601 = 682 = 512

  27. Similar right-angled triangles The following right-angled triangles are similar. 2.4 1.8 ? 2 1.5 2.5 ? 3 Find the lengths of the missing sides. Check that Pythagoras’ Theorem holds for both triangles.

  28. Similar right-angled triangles The following right-angled triangles are similar. 2.4 1.8 2 1.5 2.5 3 22 + 1.52 = 4 + 2.25 2.42 + 1.82 = 5.76 + 3.24 = 6.25 = 9 = 2.52 = 32

  29. S2 Pythagoras’ Theorem Contents S2.1 Introducing Pythagoras’ Theorem • A S2.2 Identifying right-angled triangles • A S2.3 Pythagorean triples • A S2.4 Finding unknown lengths • A S2.5 Applying Pythagoras’ Theorem in 2-D • A S2.6 Applying Pythagoras’ Theorem in 3-D • A

  30. Pythagoras’ Theorem Pythagoras’ Theorem states that for a right-angled triangle with a hypotenuse of length c and the shorter sides of lengths a and b c a c2 =a2 + b2 b We can use Pythagoras’ Theorem • to check whether a triangle is right-angled given the lengths of all the sides, • to find the length of a missing side in a right-angled triangle given the lengths of the other two sides.

  31. Finding the length of the hypotenuse Use Pythagoras’ Theorem to calculate the length of side a. a 5 cm 12 cm Using Pythagoras’ Theorem, a2 =52 + 122 a2 =25 + 144 a2 =169 a =169 a =13 cm

  32. Finding the length of the hypotenuse P Use Pythagoras’ Theorem to calculate the length of side PR. 0.7 m Q R 2.4 m Using Pythagoras’ Theorem PR2 =PQ2 + QR2 Substituting the values we have been given, PR2 =0.72 + 2.42 PR2 =0.49 + 5.76 PR2 =6.25 PR =6.25 PR =2.5 m

  33. Finding the length of the hypotenuse 6 cm Use Pythagoras’ Theorem to calculate the length of side x to 2 decimal places. x 3 cm Using Pythagoras’ Theorem x2 =62 + 32 x2 =36 + 9 x2 =45 x =45 x =6.71 cm

  34. Finding the length of the shorter sides Use Pythagoras’ Theorem to calculate the length of side a. 26 cm 10 cm a Using Pythagoras’ Theorem, a2 + 102 =262 a2 =262 – 102 a2 =676 – 100 a2 =576 a =576 a =24 cm

  35. Finding the length of the shorter sides A Use Pythagoras’ Theorem to calculate the length of side AC to 2 decimal places. 5 cm C B 8 cm Using Pythagoras’ Theorem AB2 + AC2 =BC2 Substituting the values we have been given, 52 + AC2 =82 AC2 =82 – 52 PR2 =39 PR =39 PR =6.24 cm

  36. Finding the length of the shorter sides Use Pythagoras’ Theorem to calculate the length of side x to 2 decimal places. 15 cm 7 cm x Using Pythagoras’ Theorem, x2 + 72 =152 x2 =152 – 72 x2 =225 – 49 x2 =176 x =176 x =13.27 cm

  37. Find the correct equation

  38. Complete this table

  39. S2 Pythagoras’ Theorem Contents S2.1 Introducing Pythagoras’ Theorem • A S2.2 Identifying right-angled triangles • A S2.3 Pythagorean triples • A S2.5 Applying Pythagoras’ Theorem in 2-D S2.4 Finding unknown lengths • A • A S2.6 Applying Pythagoras’ Theorem in 3-D • A

  40. Finding the lengths of diagonals Use Pythagoras’ Theorem to calculate the length of the diagonal, d. d 10.2 cm 13.6 cm Pythagoras’ Theorem has many applications. For example, we can use it to find the length of the diagonal of a rectangle given the lengths of the sides. d2 =10.22 + 13.62 d2 = 104.04 + 184.96 d2 =289 d =289 d =17 cm

  41. Finding the lengths of diagonals Use Pythagoras’ Theorem to calculate the length d of the diagonal in a square of side length 7 cm. d 7 cm Using Pythagoras’ Theorem d2 = 72 + 72 d2 =49 + 49 d2 =98 d =98 d =9.90 cm (to 2 d.p.)

  42. Finding the lengths of diagonals d =2a2 d =2 a Use this formula to find the length of the diagonal in a square with side length 12 cm to 2 decimal places. Show that the length of the diagonal d in a square of side length a can be found by the formula d = √2 a d a Using Pythagoras’ Theorem d2 = a2 + a2 d2 =2a2

  43. Finding the height of an isosceles triangle 5.8 cm h 4 cm Use Pythagoras’ Theorem to calculate the height h of this isosceles triangle. 5.8 cm h 8 cm Using Pythagoras’ Theorem in half of the isosceles triangle, we have h2 + 42 = 5.82 h2 =5.82 – 42 h2 = 33.64 – 16 h2 =17.64 h =17.64 h =4.2 cm

  44. Finding the height of an equilateral triangle 4 cm h 2 cm Use Pythagoras’ Theorem to calculate the height h of an equilateral triangle with side length 4 cm. 4 cm h Using Pythagoras’ Theorem in half of the equilateral triangle, we have h2 + 22 = 42 h2 =42 – 22 h2 =16 – 4 h2 =12 h =12 h =3.46 cm (to 2 d.p.)

  45. Finding the height of an equilateral triangle Show that the height h in an equilateral triangle of side length a can be found by the formula h = √3 a a h 2 a2 a2 h2 + = a2 We can think of this as 1a2 – a2 4 4 h2 = a2 – a h 3a2 h2 = a 4 1 2 4 3 h =a 2 Using Pythagoras’ Theorem in half of the equilateral triangle, we have

  46. Ladder problem

  47. Flight path problem

  48. Applying Pythagoras’ Theorem twice 18 cm a 4 cm 9 cm Sometimes we have to apply Pythagoras’ Theorem twice to find a required length. For example, Find the length of side a. To solve this problem we need to find the length of the other missing side which we can call b. b2 = 182 – (4 + 9)2 b2 = 324 – 169 b b2 = 155 Now, a2 = b2 + 42 a2 = 155 + 16 a2 = 171 a = 13.08 cm (to 2 d.p.)

  49. Find the given lengths y A(–4, 5) 5 4 3 2 1 x –6 –5 –4 –3 –2 –1 0 1 2 3 4 5 6 –1 –2 B(–4, –2) C (6, –2) –3 –4 –5 The points A(–4, 5), B(–4, –2), and C (6,–2) form a right angled triangle. 7 units What is length of the line AB? AB = 5 – –2 = 5 + 2 = 7 units

  50. Find the given lengths y A(–4, 5) 5 4 3 2 1 x –6 –5 –4 –3 –2 –1 0 1 2 3 4 5 6 –1 –2 B(–4, –2) C (6, –2) –3 –4 –5 The points A(–4, 5), B(–4, –2), and C (6,–2) form a right angled triangle. 7 units What is length of the line BC? 10 units BC = 6 – –4 = 6 + 4 = 10 units

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