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Subject: "pythagorean triples"     Previous Topic | Next Topic
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h20skiier4life
guest
Mar-20-02, 09:48 PM (EST)
 
"pythagorean triples"
 
   i have a hw problem for my geom. class. it'states:
1)list the pythagorean triples generated using <= 5
2)Find expressions for m, and n in terms of a, b, and c.
3)If you are given 3 numbers and asked to find the corresponding values of m and n, how can you decide which number is a, b, and c?
4)find values of m and n for the pythagorean triple 56, 90, 106.
5)find values of m and n for pythagorean triple 48, 55, 73.

i have no clue where to even start. if anyone can help me, it would be greatly appreciated : whiteknite13@hotmail.com

a = n^2 - m^2
b = 2mn
c = n^2 + m^2


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Vladimir
Member since Jun-22-03
Sep-05-03, 08:40 PM (EST)
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1. "RE: pythagorean triples"
In response to message #0
 
   LAST EDITED ON Sep-06-03 AT 02:30 AM (EST)
 
List the pythagorean triples generated using <= 5.

a = n2 - m2
b = 2穘穖
c = n2 + m2

Since a > 0, it must be n > m. Since b is even, even a at the same time would lead to a Pythagorean triple that is a multiple of some lower Pythagorean triple and we will skip those. Consequently, n and m cannot be both even or both odd.

n = 2, m = 1
a = 22 - 12 = 4 - 1 = 3
b = 2򈭽 = 4
c = 22 + 12 = 4 + 1 = 5

n = 3, m = 2
a = 32 - 22 = 9 - 4 = 5
b = 2򉁪 = 12
c = 32 + 22 = 9 + 4 = 13

n = 4, m = 1
a = 42 - 12 = 16 - 1 = 15
b = 2򉕕 = 8
c = 42 + 12 = 16 + 1 = 17

n = 4, m = 3
a = 42 - 32 = 16 - 9 = 7
b = 2򉕗 = 24
c = 42 + 32 = 16 + 9 = 25

n = 5, m = 2
a = 52 - 22 = 25 - 4 = 21
b = 2򉩂 = 20
c = 52 + 22 = 25 + 4 = 29

n = 5, m = 4
a = 52 - 42 = 25 - 16 = 9
b = 2򉩄 = 40
c = 52 + 42 = 25 + 16 = 41
---------------------------------------------------------------------

Find expressions for m, and n in terms of a, b, and c.

c + a = 2n2
c - a = 2m2

n = {(c + a)/2}
m = {(c - a)/2}

Alternately:

c + b = (n + m)2
c - b = (n - m)2

(c + b) = n + m
(c - b) = n - m

n = 1/2穥(c + b) + (c - b)}
m = 1/2穥(c + b) - (c - b)}
---------------------------------------------------------------------

If you are given 3 numbers and asked to find the corresponding values of m and n, how can you decide which number is a, b, and c?

Since c > a and c > b, c is the largest integer of the triple.
Since (c + a)/2, (c - a)/2, c + b , and c - b are all complete squares:

(c + a)/2 = n2
(c - a)/2 = m2
c + b = (n + m)2
c - b = (n - m)2

where n and m integers, neither of c + a, c - a, (c + b)/2, or (c - b)/2 can be a complete square:

c + a = 2n2
c - a = 2m2
(c + b)/2 = (n + m)2/2
(c - b)/2 = (n - m)2/2

Try to add both the remaining numbers to c and/or to subtract both the remaining numbers from c. b is the one that makes a complete square when added to or subtracted from c.
---------------------------------------------------------------------

Find values of m and n for the Pythagorean triple 56, 90, 106.

Since 106 > 56 and 106 > 90, c = 106

106 + 56 = 162 is not a complete square
106 - 56 = 50 is not a complete square
106 + 90 = 196 = 142 is a complete square
106 - 90 = 16 = 42 is a complete square

a = 56
b = 90

n = {(c + a)/2} = {(106 + 56)/2} = (162/2) = 81 = 9
m = {(c - a)/2} = {(106 - 56)/2} = (50/2) = 25 = 5

Alternately

n = 1/2穥(c + b) + (c - b)} = 1/2穥(106 + 90) + (106 - 90)} =
= 1/2(196 + 16) = 1/2(14 + 4) = 18/2 = 9

m = 1/2穥(c + b) - (c - b)} = 1/2穥(106 + 90) - (106 - 90)} =
= 1/2(196 - 16) = 1/2(14 - 4) = 10/2 = 5

---------------------------------------------------------------------

Find values of m and n for the Pythagorean triple 48, 55, 73.

Since 73 > 48 and 73 > 55, c = 73

73 + 48 = 121 = 112 is a complete square
73 - 48 = 25 = 52 is a complete square
73 + 55 = 128 is not a complete square
73 - 55 = 18 is not a complete square

a = 55
b = 48

n = {(c + a)/2} = {(73 + 55)/2} = (128/2) = 64 = 8
m = {(c - a)/2} = {(73 - 55)/2} = (18/2) = 9 = 3

Alternately

n = 1/2穥(c + b) + (c - b)} = 1/2穥(73 + 48) + (73 - 48)} =
= 1/2(121 + 25) = 1/2(11 + 5) = 16/2 = 8

m = 1/2穥(c + b) - (c - b)} = 1/2穥(73 + 48) - (73 - 48)} =
= 1/2(121 - 25) = 1/2(11 - 5) = 6/2 = 3


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