Following completion of your readings, complete the exercise 4 in the “Projects”
ID: 3102121 • Letter: F
Question
Following completion of your readings, complete the exercise 4 in the “Projects” section on page 522 of Mathematics in Our World.Make sure you build or generate at least five more Pythagorean Triples using one of the many formulas available online for doing this.
After building your triples, verify each of them in the Pythagorean Theorem equation.
The assignment must include (a) all math work required to answer the problems as well as (b) introduction and conclusion paragraphs.
Your introduction should include three to five sentences of general information about the topic at hand.
The body must contain a restatement of the problems and all math work, including the steps and formulas used to solve the problems.
Your conclusion must comprise a summary of the problems and the reason you selected a particular method to solve them. It would also be appropriate to include a statement as to what you learned and how you will apply the knowledge gained in this exercise to real-world situations.
Explanation / Answer
You can generate Pythagorean triples using the following expressions:
Pick two positive integers, m and n, with m less than n.
Then the three numbers that form the Pythagorean triple can be calculated from:
n² - m²
2mn
n² + m²
Examples:
1) m = 3, n = 4
n² - m² = (4)² - (3)² = 16 - 9 = 7
2mn = 2(3)(4) = 24
n² + m² = (4)² + (3)² = 16 + 9 = 25
Triple: 7, 24, 25
Check:
(7)² + (24)² = (25)²
49 + 576 = 625
625 = 625
2) m = 1, n = 3
n² - m² = (3)² - (1)² = 9 - 1 = 8
2mn = 2(1)(3) = 6
n² + m² = (3)² + (1)² = 9 + 1 = 10
Triple: 6, 8, 10
Check:
(6)² + (8)² = (10)²
36 + 64 = 100
100 = 100
3) m = 4, n = 5
n² - m² = (5)² - (4)² = 25 - 16 = 9
2mn = 2(4)(5) = 40
n² + m² = (5)² + (4)² = 25 + 16 = 41
Triple: 9, 40, 41
Check:
(9)² + (40)² = (41)²
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1681 = 1681
4) m = 5, n = 6
n² - m² = (6)² - (5)² = 36 - 25 = 11
2mn = 2(5)(6) = 60
n² + m² = (6)² + (5)² = 36 + 25 = 61
Triple: 11, 60, 61
Check:
(11)² + (60)² = (61)²
121 + 3600 = 3721
3721 = 3721
5) m = 2, n = 4
n² - m² = (4)² - (2)² = 16 - 4 = 12
2mn = 2(2)(4) = 16
n² + m² = (4)² + (2)² = 16 + 4 = 20
Triple: 12, 16, 20
Check:
(12)² + (16)² = (20)²
144 + 256 = 400
400 = 400