Brain Teasers
Same Hypotenuse
What is the smallest integer
that can be the hypotenuse of two different right triangles, each of which has
legs whose lengths are also integers?
This brainteaser was
written by Derrick Niederman.
Solution:
2
5 units.
The following table shows some Pythagorean triples and
several of their multiples:
|
Pythagorean Triple
|
Multiples
|
|
(3, 4, 5)
|
(6, 8, 10)
(9, 12, 15)
(12, 16, 20)
(15, 20, 25)
|
|
(5, 12, 13)
|
(10, 24, 26)
(15, 36, 39)
(20, 48, 52)
|
|
(7, 24, 25)
|
(14, 48, 50)
(21, 72, 75)
|
|
(9, 40, 41)
|
(18, 80, 82)
|
Notice that the fourth multiple of (3, 4, 5) is (15, 20,
25), which has a hypotenuse of 25 units. Similarly, the primitive triple
(7, 24, 25) also has a hypotenuse of 25 units.
There are several methods for generating Pythagorean
triples. One is to take any odd number, square it, and represent it as the sum
of consecutive integers. For instance, 72 = 49, and 49 = 24 + 25, so
(7, 24, 25) is a Pythagorean triple.
Algebraically, if the hypotenuse is k + 1 units and the longer leg is k units, then the shorter leg will be sqrt(2k+1) units. The following shows that these values satisfy
the Pythagorean Theorem:
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