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Starter. L.O. To be able to Find the missing side of a right-angled triangle, given the other two sides Solve problems using Pythagoras’ theorem, such as finding the diagonal of a rectangle Solve problems using Pythagoras’ theorem in 3 dimensions Key Words

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  1. Starter L.O. To be able to Find the missing side of a right-angled triangle, given the other two sides Solve problems using Pythagoras’ theorem, such as finding the diagonal of a rectangle Solve problems using Pythagoras’ theorem in 3 dimensions Key Words hypotenuse, right angle, square, square root, theorem

  2. CONTENTS – Click to go to… INTRODUCTION FINDING THE HYPOTENUSE FINDING ANY SIDE PROBLEM SOLVING PYTHAGORAS IN 3D REVISION

  3. INTRODUCTION

  4. SQUARES AND SQUARE ROOTS On your mini whiteboards… Quick Questions!

  5. Evaluate : 72

  6. x Square Area = 49 m2

  7. 5 m Square Area = ?

  8. Evaluate : 81

  9. x Square Area = 144 m2

  10. 10 m Square Area = ?

  11. Evaluate : 12

  12. Evaluate : 400

  13. Evaluate : 251/2

  14. Evaluate : 1

  15. Identifying the hypotenuse A right-angled triangle contains a right angle The side opposite the right angle is always the LONGEST It has a special name - HYPOTENUSE hypotenuse

  16. Which side is the hypotenuse? f e a b d c g h i

  17. Discovery task! Make accurate constructions of the three triangles below MEASURE accurately the missing length on each one. Copy and fill in this table… Can you spot the pattern that links the last three columns? 3cm 6cm 4cm 5cm 12cm 8cm

  18. Pythagoras proof activity From your worksheet… Colour in square 5 one colour, then cut it off Colour in the square with pieces 1-4 a different colour, hen cut it off, also cutting along the dotted lines to get four shapes from that square Try to arrange the five pieces you have so that they fit exactly into the larger bold square. Stick them down. Write a sentence that describes what you have discovered from this activity “I have shown that…_____________________ _____________________________________ _____________________________________”

  19. Pythagoras’ Theorem 2 2 2 a + b = h h a b

  20. c b a X X X a2 X c2 X X b2 X X + = a2 b2 c2

  21. Pythagorean trees! – Do you think you could draw one?!

  22. FINDING THE HYPOTENUSE

  23. Pythagoras’ Theorem – ALGEBRAIC METHOD Learn the formula! Substitute the two values that you know into the formula Solve the equation to find the unknown value 2 2 2 a + b = h h a b

  24. 1 2 x x 5 cm 4.6 cm 9.8 cm 12 cm Example – Algebraic method

  25. Pythagoras – QUICK METHOD Decide if the side you are LOOKING FOR is the HYPOTENUSE, or one of the two SHORTER SIDES Use one of these quick and easy methods to work out what you need • Looking for the LONGEST SIDE? • (the hypotenuse!) • Square it • Square it • Add it • Square root it • Looking for a SHORTER SIDE? • (not the hypotenuse!) • Square it • Square it • Add it • Square root it

  26. 2 1 x x 3 cm 4.6 cm 4 cm 9.8 cm Example – Quick method Square it… Square it… Add it… Square root it… Square it… Square it… Add it… Square root it… (1dp)

  27. Task – Find the missing side, to 2d.p. 10 cm a b 1) b = 15.62cm 2) 4 cm 6 cm 12 cm a = 7.21cm 5.3 cm 5 cm 8.1 cm c 3) 4) d d = 9.68cm c = 7.21cm 7cm 10.1 cm 12 cm 8.5 cm 9.7 cm 5) 6) f = 15.43cm e = 8.60cm e f

  28. Task – Find the missing side, to 2d.p. 13.04 m 16.12 cm 1) 3) 2) x x 8 cm 9.85 m 9 m 7 m x 11 m 4 m 14 cm 4) 5) 6) 6 m x x 16.46 mm 9.76 m 7.81 m 5 m 5.3 m 8.2 m x 14.7 mm 7.4 mm

  29. FINDING ANY SIDE

  30. Pythagoras’ Theorem – ALGEBRAIC METHOD Learn the formula! Substitute the two values that you know into the formula Solve the equation to find the unknown value 2 2 2 a + b = h h a b

  31. x m 25 m 7 m Example – Algebraic method

  32. Pythagoras – QUICK METHOD Decide if the side you are LOOKING FOR is the HYPOTENUSE, or one of the two SHORTER SIDES Use one of these quick and easy methods to work out what you need • Looking for the LONGEST SIDE? • (the hypotenuse!) • Square it • Square it • Add it • Square root it • Looking for a SHORTER SIDE? • (not the hypotenuse!) • Square it • Square it • Add it • Square root it

  33. 2 1 7cm 15cm x x 9cm 5cm Example – Quick method Square it… Square it… 81 Subtract it… Square root it… Square it… Square it… Subtract it… Square root it… (1dp)

  34. Task – Find the missing side, to 2d.p. 5.2 cm 10 cm 12.4 cm 2) 1) h = 11.26cm 4 cm g h g = 9.17cm 4) j 3) 14.5 cm 8.4 cm i i = 4.45cm j = 23.86cm 25.3 cm 13.8cm 2.9 cm 23 cm 6) l 5) k k = 9.15cm l = 19.26cm 30 cm 9.6 cm

  35. Task – Find the missing sides, to 2.d.p 12.69 m 3) 1) 2) x 12 m 15 m x 8 m 8 mm 9.75 m 24 mm 22.63 mm x 7 m 4) 5) 6) 52 mm 14 m 18 m 28 mm x 20 m x 43.82 mm 5 m 17.29 m 14.28 m x

  36. Task – Find the missing sides, to 2.d.p 5.3 3) 1) 2) 10.44 34 x x 27.50 9 14.32 6 x 20 13 4) 5) 6) x 1.8 18.25 4.48 14 9.80 x 23 x 10 14 4.1

  37. Exam question – Spot the errors! a) FIND PQ 361 + 225 So PQ is 586cm b) FIND THE AREA OF PQR

  38. PYTHAGORAS CATCH PHRASE! On your mini whiteboards… Choose a question to answer. Get it right to reveal part of the picture and guess the catchphrase!

  39. 6.71cm The diagram shows a trapezium ABCD . Calculate the length of AB. 12.73m 5cm This rectangle has a diagonal of 13cm and a width of 12cm. Calculate it’s height. 8cm C A c 9m 13cm 7cm j B D 9m 10cm 12cm 151.99 miles 16.52cm 8.54m Cardiff is 130 miles west of London. Hull is 200 miles north east of Cardiff. How far is Hull from London? Hull 23cm 8m d 200 miles 16cm London Cardiff 3m 130 miles f 13.23cm 6.93m 8.94cm b k 20cm a 4m 8cm 12cm 8m 15cm

  40. BAD HAIR DAY!

  41. PROBLEM SOLVING

  42. Problem solving with Pythagoras Draw yourself a diagram which includes a right-angled triangle Identify which side is the hypotenuse Use Pythagoras in the normal way to find the missing side

  43. 1 Find the diagonal of the rectangle 6 cm d 9.3 cm 2 A rectangle has a width of 4.3 cm and a diagonal of 7.8 cm. Find its perimeter. 7.8 cm 4.3 cm x cm Examples Perimeter = 2(6.5+4.3) = 21.6 cm

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