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Pythagoras’ Theorem

Pythagoras’ Theorem. Only works in right angled triangles Nothing to do with angles. Hypotenuse. The hypotenuse is the longest side in a right angled triangle. It is always the side opposite the right angle. hypotenuse. Spotting the Hypotenuse. hypotenuse. Pythagoras’ Theorem.

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Pythagoras’ Theorem

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  1. Pythagoras’ Theorem • Only works in right angled triangles • Nothing to do with angles

  2. Hypotenuse • The hypotenuse is the longest side in a right angled triangle. • It is always the side opposite the right angle. hypotenuse

  3. Spotting the Hypotenuse hypotenuse

  4. Pythagoras’ Theorem • The area of the square drawn on the hypotenuse is equal to the sum of the area of the squares drawn on the other two sides • c2 = a2 + b2 c a b

  5. Pythagoras’ Theorem You can visualise the theorem a c b c2= a2 + b2

  6. Finding the hypotenuse Find the missing side. 100 + 324 = 424 c2 = 102 + 182 c2 = 424 100 10 ? 18 c = √424 324 c = 20.6 (1.d.p.)

  7. 24km 5km 11m 7m Find the missing sides Give your answers to 1 d.p. 8cm 10cm c2 = a2 + b2 c2 = 242 + 52 c2 = 576 + 25 c2 = 601 c = √601 c = 24.5km (1.d.p.) c2 = a2 + b2 c2 = 102 + 82 c2 = 100 + 64 c2 = 164 c = √164 c = 12.8cm (1.d.p.) c2 = a2 + b2 c2 = 112 + 72 c2 = 121 + 49 c2 = 170 c = √170 c = 13.0m (1.d.p.)

  8. Finding a shorter side Find the missing side. 182 = a2 + 102 324 a2 = 182 - 102 18 a2 = 224 100 10 ? a = √224 324 – 100 = 224 a = 15.0 (1.d.p.)

  9. 24km 5km 17m 11m Find the missing sides Give your answers to 1 d.p. 20 cm 12 cm a2 = c2 - b2 a2 = 242 - 52 a2 = 576 - 25 a2 = 551 a = √551 a = 23.5km (1.d.p.) a2 = c2 - b2 a2 = 202 - 122 a2 = 400 - 144 a2 = 256 a = √256 a = 16 cm a2 = c2 - b2 a2 = 172 - 112 a2 = 289 - 121 a2 = 168 a = √168 a = 13.0m (1.d.p.)

  10. N W E S Navigation Navigation problems are often solved using Pythagoras’ Theorem.

  11. ? 41km 32km Airport Navigation A plane leaves an airport and travels 32km west then it turns and travels 41km north. It develops a problem and has to return to the airport. How far is it? Step 1. Draw a diagram Step 2. Use Pythagoras c2 = a2 + b2 c2 = 322 + 412 c2 = 1024 + 1681 c2 = 2705 c = √2705 c = 52.0km (1.d.p.)

  12. Isosceles Triangle Problems involving isosceles triangles are often solved using Pythagoras’ Theorem.

  13. Isosceles Triangle c2 = a2 + b2 • Draw perpendicular and mark lengths • Use Pythagoras theorem c a b b

  14. Isosceles Triangle A roof on a house that is 6 m wide peaks at a height of 3 m above the top of the walls. Find the length of the sloping sides of the roof. Step 1. Draw a diagram c 3 m Step 2. Use Pythagoras ? ? 3 m 3 m c2 = a2 + b2 c2 = 32 + 32 c2 = 9 + 9 c2 =18 c = √18 c = 4.2 m (1.d.p.) 6 m

  15. Solving Problems Using Pythagoras’ Theorem • draw a diagram for the problem that includes a right-angled triangle • label the triangle with the length of its sides from the question • label the unknown side ‘x’ • if it’s the hypotenuse, then • “SQUARE, SQUARE, ADD, SQUARE ROOT” • if it’s one of the shorter sides, then • “SQUARE, SQUARE, SUBTRACT, SQUARE ROOT” • round your answer to a suitable degree of accuracy

  16. Solving Problems Using Pythagoras’ Theorem Is this triangle possible?

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