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Work out missing lengths for squares and cuboids

Learn how to calculate missing side lengths in squares and cuboids using Pythagoras' Theorem. This lesson covers right-angled triangles, identifying hypotenuse, and applying the theorem in 2D and 3D scenarios.

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Work out missing lengths for squares and cuboids

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  1. Work out the missing lengths for these squares and cuboids Starter Area = 25cm2 Area = 64cm2 Area = 81 cm2 x x x Area = 6.25cm2 Vol = 64cm3 x y Vol = 216cm3 y

  2. KS4 Mathematics S2 Pythagoras’ Theorem

  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. Objective: • By the end of the lesson you should be able to work out the missing side of any right angled triangle using Pythagoras’ Theorem. • Key words: • Hypotenuse • Pythagoras’ theorem • Right angled triangle • Big blue Foundation books • Pages 459-460 • For the brave pages 461-463 • PLEASE DRAW THE TRIANGLE AND SHOW YOUR WORKING CLEARLY ALWAYS THINK ARE YOU TRYING TO FIND A LONGER SIDE OR A SHORTER SIDE

  6. 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.

  7. 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.

  8. Showing Pythagoras’ Theorem

  9. 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

  10. 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

  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. 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

  13. 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

  14. 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

  15. 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

  16. 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

  17. 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

  18. 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

  19. 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.)

  20. 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

  21. 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.)

  22. Ladder problem

  23. Flight path problem

  24. 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

  25. 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.

  26. 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.

  27. 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.

  28. 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.

  29. 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

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