Unit 7A - Lesson #4: Pythagorean Theorem (Textbook Section 7-2)

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Unit 7A - Lesson #4: Pythagorean Theorem (Textbook Section 7-2) Please pick up and complete the Lesson #4: Right Triangle Investigation example sheet found on the front table. Grab a Ruler and Calculator as well.

Leg Investigation Hypotenuse 3 4 5 25 25 6 8 10 100 5 12 1 3 169 Leg1 Leg1 Leg2 Hypotenuse (Leg1)2 + (Leg2)2 Hypotenuse2 Triangle #1 3 4 5 25 25 Triangle #2 6 8 10 100 Triangle #3 5 12 1 3 169 Leg Hypotenuse

c a b Pythagorean Theorem In any right triangle with leg lengths of a and b, and a hypotenuse with length c, then a2 + b2 = c2. a b c

Pythagorean Triples Sets of three whole numbers that are known to form a right triangle. Famous Triples 3, 4 ,5 5, 12, 13 8, 15, 17 7, 24, 25 Doubled 6, 8, 10 10, 24, 26 16, 30, 34 14, 48, 50 Tripled 9, 12, 15 15, 36, 39 24, 45, 51 21, 72, 75

Example #1: Determining Right Triangles No Yes (A) 3, 7, 8 _______ (B) 10, 24, 26 _______ 32 + 72  82 9 + 49  64 58  64 102 + 242 = 262 100 + 576 = 676 676 = 676 Power of Pythagorean Triples: (5, 12, 13)  2 = 10, 24, 26

Example #1: Determining Right Triangles Yes No (C) _______ (D) 6, 11, 9 _______ 62 + 92  112 36 + 81  121 117  121

Using the Pythagorean Theorem Finding the Hypotenuse 4 3 x Pythagorean Triple: 3, 4, 5

Power of Pythagorean Triples x2 + 1296 = 1521 x2 = 225 y 36 39 3 = 13 3 = 12 Pythagorean Triple? ?, 12, 13 or 12, ?, 13 5, 12, 13 3 15

Exact Answers

Examples of the Pythagorean Theorem Finding a Leg x2 + 92 = 172 x2 + 81 = 289 x2 = 208 x 9 17 (Exact Answer) (Approximate)

Example #2: Using the Pythagorean Thm. 15 (A) x = _________ (B) x = _________ x  ________ x  ________ 10.6 Not Needed 92 + 122 = x2 122 + y2 = 162 Pyth. Triple? 9  3 = 3 12  3 = 4 Look for: 3, 4, ? 3, 4, 5 3 15

Example #2: Using the Pythagorean Thm. 30 4 (C) z = _________ (D) w = _________ z  ________ w  ________ Not Needed Not Needed z2 + 162 = 342 22 + = w2 34 z 16 z2 = 900 4 + 12 = w2 16 = w2 2 w Pyth. Triple? 16  2 = 8 34  2 = 17 Look for: ?, 8, 17 or 8, ?, 17 8, 15, 17 2 30

Example #3: Right Triangles in Polygons (A) x = _________ 14 12 x w2 + 62 = 142 6 6

Example #3: Right Triangles in Polygons (B) Perimeter = _________________ 10 16 9 3 x 92 + 32 = x2

Example #4 A farmer is 5’ 7” tall. He wants to clean a spot 13 feet up on inside of his silo that has developed some mold. He currently has a 10 foot ladder. Will he be able to reach the spot with his current equipment? Write a response using complete sentences, proper spelling, grammar, and punctuation. Things to Consider: Spacing of the rungs on the ladder? How many rungs up can the farmer use? How close is the base of the ladder to the silo wall?

Example #4 Forming a right triangle with the ladder and the silo wall, we can use the Pythagorean Theorem to determine how high up the wall the ladder reaches. Adding the farmer’s height of 5’7”, it is reasonable to say that he will be able to reach the mold. Ground Silo Wall Ladder Triangle #1 0.5 9.98 10 Triangle #2 1 9.95 Triangle #3 1.5 9.89 Triangle #4 2 9.8 Triangle #5 2.5 9.68 Triangle #6 3 9.54 Triangle #7 3.5 9.37 Triangle #8 4 9.16 Triangle #9 4.5 8.93 Triangle #10 5 8.66 Triangle #11 5.5 8.35 SiloWall Ground Ladder (10 feet)