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Chapter 8: Right Triangles

Chapter 8: Right Triangles. Watch the follow clip about the Sunshine Skyway Bridge. Think about the design and construction. Do you recognize any right triangles?. Integers. Integer: One of a set of positive and negative whole numbers including zero. -3, -2, -1, 0, 1, 2, 3…

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Chapter 8: Right Triangles

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  1. Chapter 8: Right Triangles • Watch the follow clip about the Sunshine Skyway Bridge. Think about the design and construction. Do you recognize any right triangles?

  2. Integers • Integer: One of a set of positive and negative whole numbers including zero. • -3, -2, -1, 0, 1, 2, 3… • When using measurements: • Positive Integers Take a look at three squares: 9, 16, and 25. Note that 9+16=25, or 32 + 42 = 52

  3. Manipulating equations • 32 + 42 = 52 Is an equation. • Can we find more numbers among the squared integers such that the sum of the two smaller squares is equal to the largest square? • Because we have an equation, we can simply multiply each side of the equation by the same number to get another equation. • 32 + 42 = 52 • (3*2)2 +(4*2)2 = (5*2)2 62 + 82 = 102 36 + 64 = 100 TRUE

  4. Pythagorean Triples • Integers a, b, and c form a Pythagorean Triple if a2 + b2 = c2, where a and b are the smaller numbers and c is the largest. • Take 5, 12, and 13. How can we tell if they are Pythagorean Triples? PLUG IT IN!  • Does 52 + 122 = 132 ? • 25+144= 169 ? • 169 = 169 YES, therefore they are Triples. • Remember, more triples can be created by multiplying each integer in the equation by the same number.

  5. Pythagorean triples • Is (4, 5, 6) a Pythagorean Triple? • 4 and 5 are the smaller integers; 6 is the largest • 42 + 52 = 62 • 16+25=36 FALSE. • (4, 5, 6) is NOT a Pythagorean Triple.

  6. Practice • Take out personal whiteboards and determine whether each set of integers provided is a Pythagorean Theorem. • Keep notes out for later.

  7. Plato’s formula Plato provided us with many awesome ideas. One of them is a way to mathematically calculate many of the Pythagorean Triples. Don’t you wish you could have hung out with him? For any positive integer, m: (2m)2 + (m2– 1)2 = (m2 +1)2

  8. Plato’s Formula (2m)2 + (m2 – 1)2 = (m2 +1)2 • Use Plato’s Formula for m=2. Check to see if your answer is a Pythagorean Triple. • (2*2)2 + (22 – 1)2 = (22 +1)2 • (4)2 + (3)2 = (5)2 • 16+9=25 TRUE. • Plato’s Formula shows that (3,4,5) is a Pythagorean Triple.

  9. Practice (you know you love it) • Use Plato’s formula to find Pythagorean Triples for the following integers. Use a calculator if necessary. • m=6 (display on whiteboard when done) • Answer: (12, 35, 37) • m=3 (display on whiteboard when done) • Answer: (6, 8, 10)

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