The Pythagorean Theorem

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The Pythagorean Theorem

  1. 1. The Pythagorean Theorem<br />
  2. 2. The Pythagorean Theorem<br />Named for Pythagoras, a Greek mathematician who lived in the sixth century B.C. <br />Babylonians, Egyptians, and Chinese were aware of this relationship<br />
  3. 3. The Pythagorean Theorem<br />In a right triangle, the side opposite the right angle is the longest side. It is the hypotenuse. The other two sides are the legs of a right triangle.<br />hypotenuse<br />legs<br />
  4. 4. In a right triangle, the sum of the squares of the lengths of the legs is equal to the square of the length of the hypotenuse.<br />a2+ b2= c2<br />The Pythagorean Theorem<br />c<br />a<br />b<br />
  5. 5. Try these:<br />A right triangle has sides of lengths 20, 29, and 21. What is the length of the hypotenuse?<br />Verify that the Pythagorean Theorem is true for the right triangle in the previous question.<br />Find the length of the hypotenuse of a right triangle with legs of lengths 7 and 24.<br />WHO? Czech-American mathematician Olga Taussky-Todd (1906-1995) studied Pythagorean triangles. In 1970, she won the Ford Prize for her research<br />
  6. 6. Sometimes you will leave your answer in simplest radical form.<br />Find the value of x. Leave your answer in simplest radical form.<br />The hypotenuse of a right triangle has length 12. one leg has length 6. Find the length of the other leg in simplest radical form.<br />20<br />8<br />x<br />
  7. 7. Find the value of x.<br />1. 2.<br />3. 4.<br />26<br />26<br />x<br />x<br />6<br /> 48<br />8<br />x<br />3<br />8<br />x<br />x<br />x<br />2<br />3<br />x<br />
  8. 8. When the lengths of the sides of a right triangle are integers, the integers form a Pythagorean Theorem. Here are some common primitive Pythagorean Triples.<br />3, 4, 5<br />5, 12, 13<br />8, 15, 17<br />7, 24, 25<br /><ul><li>9, 40, 41
  9. 9. 11, 60, 61
  10. 10. 12, 35, 37
  11. 11. 13, 84, 85</li></ul>Choose an integer. Multiply each number of a Pythagorean triple by that integer. Verify that the result is a Pythagorean triple.<br />
  12. 12. What is the length of the diagonal of a rectangle whose sides measures 5 and 7?<br />Calculate the length of the side of a square whose diagonal measures 9 cm.<br />What is the measure of the longest stick we can put inside a 3 cm x 4 cm x 5 cm box?<br />Pythagorean Theorem & Other Shapes<br />
  13. 13. Pythagorean Theorem Converse<br />In ΔABC with longest side c,<br />if c2 = a2 + b2, then the triangle is right.<br />if c2 > a2 + b2, then the triangle is obtuse.<br />if c2 < a2 + b2, then the triangle is acute.<br />B<br />c<br />a<br />C<br />A<br />b<br />
  14. 14. The number represent the lengths of the sides of a triangle (a, b, c). Classify each triangle as acute, obtuse, or right.<br />obtuse<br />right<br />acute<br />right<br />obtuse<br />right<br />acute/equi<br />acute<br />right<br />right<br />2, 3, 4<br />3, 4, 5<br />4, 5, 6<br />3, 3, 32<br />3, 3, 33<br />2, 23, 4<br />5, 5, 5<br />4, 4, 5<br />2, 2, 2<br />2.5, 6, 6.5<br /> <br />
  15. 15. The number represent the lengths of the sides of a triangle (a, b, c). Classify each triangle as acute, obtuse, or right.<br />obtuse<br />right<br />acute<br />right<br />obtuse<br />2, 3, 4<br />3, 4, 5<br />4, 5, 6<br />3, 3, 32<br />3, 3, 33<br />2, 23, 4<br />5, 5, 5<br />4, 4, 5<br />2, 2, 2<br />2.5, 6, 6.5<br /> <br />

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