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The Pythagorean Theorem

The Pythagorean Theorem. Pythagoras and his Theorem. Right Triangle : a triangle with exactly one right angle. Legs : the sides of a triangle that form the right angle. Hypotenuse : the longest side, its located across from the right angle. c a b.

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The Pythagorean Theorem

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  1. The Pythagorean Theorem

  2. Pythagoras and his Theorem • Right Triangle: a triangle with exactly one right angle. • Legs: the sides of a triangle that form the right angle. • Hypotenuse: the longest side, its located across from the right angle.

  3. c a b The legs are labeled a & b and the hypotenuse is ALWAYS labeled c.

  4. Pythagoras discovered that the sum of the squares of the two legs in a right triangle is equal to the square of the hypotenuse. That means, in any right triangle, a² + b² = c² leg² + leg² = hypotenuse²

  5. The term “squared” comes from the area of a square. EX: 3 “squared” means 3x3 or 9. The area of a 3x3 square is 9

  6. Could a right triangle have sides that measure 3 cm, 4 cm, and 5 cm? a² + b² = c² 3² + 4² ? 5² 9 + 16 ? 25 25 = 25 Yes, this is a right triangle because the Pythagorean Theorem works!

  7. How about sides of 5, 6, and 7? a² + b² = c² 5² + 6² ? 7² 25 + 36 ? 49 61 ≠ 49 NO, this is a NOT right triangle because the Pythagorean Theorem doesn’t work!

  8. Is 15, 17, 8 a right triangle? Why or why not? Show Work! a² + b² = c² 8² + 15² ? 17² 64 + 225 ? 289 289 = 289 Yes, this is a right triangle because the Pythagorean Theorem works!

  9. Using the Pythagorean Theorem to Find a Missing Side a² + b² = c² 5² + 12² = c² 25 + 144 = c² 169 = c² √169 = √c² 13 = c Note: the missing side is the hypotenuse

  10. What if you know the hypotenuse? You can use the theorem to find one of the legs. a² + b² = c² 9² + b² = 15² 81 + b² = 225 -81 -81 b² = 144 √b² = √144 b = 12

  11. When your answers don’t work out evenly, round to the nearest TENTH . a² + b² = c² a² + 4² = 11² a² + 16 = 121 -16 -16 b² = 105 √b² = √105 b = 10.246 10.2m

  12. Finally, we can use the Pythagorean Theorem to solve real life word problems. Jen hiked 8 miles east, then turned and hiked 6 miles south. How far was she from her starting point? DRAW A PICTURE!

  13. Jen hiked 8 miles east, then turned and hiked 6 miles south. How far was she from her starting point? 8 miles east 6 miles south ? a² + b² = c² 8² + 6² = c² 64 + 36 = c² 100 = c² √100 = √c² 10 = c Jen was 10 miles from where she started.

  14. Polygons • Polygon: a closed figure formed by 3 or more line segments that intersect only at their verticies. • Polygons are classified by the number of sides and angles they have

  15. Polygons 3 sides: triangle 4 sides: quadrilateral 5 sides: pentagon 6 sides: hexagon 9 sides: nonagon 10 sides: decagon 7 sides: heptagon 8 sides: octagon

  16. Regular Polygons • Regular Polygon: a polygon in which all the sides are the same length and all the angles are the same measure. • Example:

  17. Interior Angles • What if we wanted to know the measure of EACH interior angle of a regular pentagon? How would we go about doing this? • Discuss with your partner. Sum of Interior Angle Formula: (n – 2) * 180

  18. Find the measure of each interior angle of a pentagon. 108º

  19. Find the measure of each interior angle of a hexagon. 120º

  20. How many sides does a polygon have if the sum of its interior angles is 1440º. 10 sides

  21. Find the measure of the missing angle in the figure below. 93º

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