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Using the Pythagorean Theorem

Explore various problems using the Pythagorean Theorem to find unknown side lengths and angles in different scenarios. Discover Pythagorean triples and their applications in geometry.

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Using the Pythagorean Theorem

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  1. Using the Pythagorean Theorem

  2. Information

  3. Diagonals

  4. Ladder problem

  5. Flight path problem

  6. Using the Pythagorean theorem twice Find the length of side x. First, find the value of y2. h2 = a2 + b2 write the Pythagorean theorem: 182 = y2+ (4 + 9)2 substitute for given values: y2 = 182– (4 + 9)2 A rearrange: y2 = 155 evaluate: Use this value to find the length of x. write the Pythagorean theorem: x2 = y2 + 42 18cm y x x2 = 155 + 42 substitute: x2 = 171 evaluate: x = 13.08cm (nearest hundredth) square root both sides: B C D 4cm 9cm

  7. Pencil box This is Maria’s new pencil case. What is the longest pencil that can fit inside (diagonally) when the lid is on? The longest diagonal is y. apply the Pythagorean theorem to find x2: x2 = 42 + 22 x2 = 20 evaluate: apply the Pythagorean theorem to find y: y2 =182 + x2 y2 = 182 + 20 substitute: y2 = 344 evaluate: y = 18.54cm (nearest hundredth) square root both sides of the equation: The longest pencil must be less than 18.54cm long.

  8. Pythagorean triples A triangle has sides of length 3cm, 4cm and 5cm. Does this triangle have a right angle? The Pythagorean theorem states 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. If the triangle has a right angle, then 32 + 42 will be equal to 52 (this is the Pythagorean theorem) evaluating: 32 + 42 = 9 + 16 52 = 5 × 5 = 25 = 25 Therefore the triangle has a right angle. The numbers 3, 4 and 5 form a Pythagorean triple.

  9. Pythagorean triples 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. 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.

  10. Pythagorean triples How many Pythagorean triples 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.

  11. Finding the height of triangles

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