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Geometry Day 56

Geometry Day 56. The Pythagorean Theorem. Objectives:. Proving the Pythagorean Theorem The Pythagorean Theorem and its converse Pythagorean inequalities. Review. The Geometric Mean proportions in right triangles are as follows:. a. b. h. d. e. c.

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Geometry Day 56

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  1. Geometry Day 56 The Pythagorean Theorem

  2. Objectives: • Proving the Pythagorean Theorem • The Pythagorean Theorem and its converse • Pythagorean inequalities

  3. Review • The Geometric Mean proportions in right triangles are as follows: a b h d e c

  4. We can rewrite the proportions as follows: a b h d e c

  5. By the addition property of equality, we get: a b h d e c

  6. We can factor the right side: • And use substitution: a b h d e c

  7. The Pythagorean Theorem • Therefore: a b h d e c

  8. The Pythagorean Theorem • The sum of the areas of the squares whose sides are the legs of a right triangle is equal to the area of the square whose side is the hypotenuse of that triangle. • This relationship among the sides of right triangles is known as the Pythagorean Theorem in this culture. • In simplest form a2 + b2 = c2, where a and b are the lengths of the legs and c is the length of the hypotenuse.

  9. The Pythagorean Theorem • We have just proven that the Pythagorean Theorem works for any right triangle. • There are hundreds of known proofs of the Pythagorean Theorem. • See handout • Project due by March 16 (last A day before Spring Break).

  10. Pythagorean Triples • If three whole numbers fit into the Pythagorean Theorem, they are called a Pythagorean Triple. • 3, 4, 5 is the most well known Pythagorean Triple. • Multiples (scales) of this Triple also work: • 6, 8, 10 • 30, 40, 50 • etc. • Knowing a few common Triples can make problem solving easier. • p. 542 of your book lists several well-known Pythagorean Triples. • Note: your book doesn’t mention this, but scale Pythagorean Triples down works as well. • Ex: 3/5, 4/5, 5/5 (=1) make a right triangle.

  11. Pythagorean Triples • Exercise: In triples, complete the following. Each member should complete the work on his/her own paper, but each group will turn in a separate set of answers.

  12. The Converse of the Pythagorean Theorem • If a2 + b2 = c2, then the triangle is a right triangle. • If a2 + b2 > c2, then the triangle is acute. • If a2 + b2 < c2, then the triangle is obtuse. • Important Note: a, b, and c must be valid lengths to form a triangle! • What type of triangle will be formed by the following lengths?

  13. Homework 32 • Workbook, p. 100

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