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Pythagorean Theorem and its Converse. Book Section 7-2 Online Unit 1 Lessons 3 & 4. Instruction. Pythagorean Theorem: 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 a 2 + b 2 = c 2. c. a. b. Instruction.
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Pythagorean Theorem and its Converse Book Section 7-2Online Unit 1 Lessons 3 & 4
Instruction • Pythagorean Theorem: • 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 hypotenusea2 + b2 = c2 c a b
Instruction • Pythagorean Triple: • Set of nonzerowhole numbers a, b, and c that satisfy equation a2 + b2 = c2 • Examples: • 3, 4, 5 • 5, 12, 13 • 7, 24, 25 Fun Fact: If you multiply each number in a Pythagorean Triple by the same whole number, the three numbers that result also form a Pythagorean Triple.
Instruction • Converse of the Pythagorean Theorem: • If the square of the length of one side of a triangle is equal to the sum of the squares of the lengths of the other two sides, then the triangle is a right triangle.
Instruction • Theorem: • If the square of the length of the longest side of a triangle is greater than the sum of the squares of the lengths of the other two sides, the triangle is obtuseIf c2 > a2 + b2, then the triangle is obtuse
Instruction • Theorem: • If the square of the length of the longest side of a triangle is less than the sum of the squares of the lengths of the other two sides, the triangle is acuteIf c2 < a2 + b2, then the triangle is acute
Let’s Practice • Find x. • a2 + b2 = c2 • 62 + 82 = c2 • 36 + 64 = c2 • 100 = c2 • 10 = c
Let’s Practice • Find x. Leave your answer in simplest radical form • a2 + b2 = c2 • 42 + b2 = 72 • 16 + b2 = 49 • b2 = 33
Let’s Practice • Allentown is 35 miles North and 12 miles east of Philadelphia. A train runs on a straight track between the two towns. How many miles does it cover? 122 + 352 = c2 144 + 1225 = c2 35 * Allentown 1369 = c2 37 = c Phil 12
Let’s Practice • Find the area of the shaded region. A = ½ bh • a2 + b2 = c2 A = ½ (12) • 62 + b2 = 82 • 36 + b2 = 64 • b2 = 28 6 h 8
Let’s Practice • The lengths of the sides of a triangle are 15, 8, and 21. Is this a right, acute, or obtuse triangle? • If it’s a right trianlge: • a2 + b2 = c2 • 289 < 441 • 289 < c2 • 152 + 82 = 212 • 225 + 64 = 441 OBTUSE • 289 = 441
Review • Pythagorean Theorem:a2 + b2 = c2 • If c2 < a2 + b2, then the triangle is acute • If c2 > a2 + b2, then the triangle is obtuse
Follow Up • Each question is worth 1 extra credit point added to your chapter 1 test • Submit via webmail with subject heading “7-2 Extra Credit” 4. A ladder resting against the side of a building reaches 32ft up the building. If the ladder is 40ft long, how far from the building is the base of the ladder? Show work. 1. The lengths of two legs of a right triangle are 16cm and 30cm. What is the length of the hypotenuse? Show work 2. The length of the hypotenuse of a right triangle is 65 yards. The length of one leg is 25 yards. What is the length of the other leg? Show work 5. The following are the lengths of the sides of three triangles. • Which is closest to the length of the diagonal of a square whose sides measure 20 cm? Show work • 20 cm • 28cm • 40cm • 200cm Which are right triangles? Show work
