Answer:
x = 51
Explanation:
The missing value of the Pythagorean triple can be found using the Pythagorean theorem, or it can be found by comparing the values in the triple to known triples.
What are Pythagorean triples?
A Pythagorean triple is a set of integers {a, b, c} that satisfy the equation of the Pythagorean theorem:
a² +b² = c²
The smallest such triple is {3, 4, 5}. It is also the only triple that is an arithmetic sequence. Other triples of small integers are ...
{5, 12, 13}, {7, 24, 25}, {8, 15, 17}
There are an infinite number of "primitive" Pythagorean triples, ones that are not multiples of another triple.
What is this triple?
The given values of the triple have the ratio ...
68/85 = (4·17)/(5·17) = 4/5
Only the values in the {3, 4, 5} triple and its multiples will have this ratio.
The value of x is 3·17 = 51, so the triple is {51, 68, 85}.
x = 51
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Additional comments
Any primitive triple will have two odd numbers.
The ratios of numbers in a primitive triple are unique to that triple. That is, the numbers are mutually prime.
For any pair of positive integers m > n, there is a Pythagorean triple {2mn, m²-n², m²+n²}. Such triples will not be primitive if m and n have the same parity.
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The above triple can be verified using the Pythagorean theorem:
51² +68² = 2601 +4624 = 7225 = 85²