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5.4: The Pythagorean Theorem

5.4: The Pythagorean Theorem. Theorem 5.4.1: The altitude drawn to the hypotenuse of a right triangle separates the right triangle into two right triangles that are similar to each other and to the original triangle.

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5.4: The Pythagorean Theorem

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  1. 5.4: The Pythagorean Theorem • Theorem 5.4.1: The altitude drawn to the hypotenuse of a right triangle separates the right triangle into two right triangles that are similar to each other and to the original triangle. In the figure below, AD and DB are segments of the hypotenuse and AD is the segment adjacent to leg AC and DB is the segment adjacent to leg BC. Section 5.4 Nack

  2. Theorem 5.4.2: The length of the altitude to the hypotenuse of a right triangle is the geometric mean of the lengths of the segments of the hypotenuse. AD = CD CD DB Note: CD is a geometric mean because the second and the third terms are identical. • Lemma 5.4.3: The length of each leg of a right triangle is the geometric mean of the length of the hypotenuse and the length of the segment of the hypotenuse adjacent to that leg. Section 5.4 Nack

  3. Pythagorean Theorem • Theorem 5.4.4: The square of the length of the hypotenuse of a right triangle is equal to the sum of the squares of the lengths of the legs. • Proof p. 243: From diagram (b) c = b and c = a b x a y Cross multiply: b² = cx and a²= cy Use the Addition Property of Equality: a² + b² = cx + cy =c (x + y) But x + y = c so: a² + b² = c² Section 5.4 Nack

  4. Theorem 5.4.5: (Converse of Pythagorean Theorem): If a, b, and c are the lengths of the three sides of a triangle, with c the length of the longest side, and if c² = a² + b², then the triangle is a right triangle with the right angle opposite the side of length c. • Proof of HL (Theorem 5.4.6): If the hypotenuse and a leg of one right triangle are congruent to the hypotenuse and a leg of the second right triangle, then the triangles are congruent. • Use the Pythagorean Theorem p. 246 Ex. 3 p. 244 Section 5.4 Nack

  5. Pythagorean Triples • A set of three natural numbers (a, b, c) for which a² + b² = c² See Table 5.1 p. 247 • Theorem 5.4.7: Let a, b, and c represent the lengths of the three sides of a triangle, with c the length of the longest side. • If c² > a² + b² , then the triangle is obtuse and the obtuse angle lies opposite the side of length c. • If c² < a² + b² , then the triangle is acute. Ex. 6 p. 247 Section 5.4 Nack

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