Survey
* Your assessment is very important for improving the workof artificial intelligence, which forms the content of this project
* Your assessment is very important for improving the workof artificial intelligence, which forms the content of this project
Line (geometry) wikipedia , lookup
John Wallis wikipedia , lookup
Proofs of Fermat's little theorem wikipedia , lookup
Abuse of notation wikipedia , lookup
Large numbers wikipedia , lookup
Collatz conjecture wikipedia , lookup
Elementary algebra wikipedia , lookup
Elementary mathematics wikipedia , lookup
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… • When using measurements: • Positive Integers N: 1 2 3 4 5 6 7 8 9 N2 1 4 9 16 25 36 49 64 81 Take a look at three squares: 9, 16, and 25. Note that 9+16=25, or 32 + 42 = 52 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 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. PYTHAGOREAN TRIPLES • Is (4, 5, 6) a Pythagorean Triple? • • • • 4 and 5 are the smaller integers; 6 is the largest 42 + 5 2 = 62 16+25=36 FALSE. (4, 5, 6) is NOT a Pythagorean Triple. PRACTICE • Take out personal whiteboards and determine whether each set of integers provided is a Pythagorean Theorem. • Keep notes out for later. 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 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. 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)