Advanced Geometry
Mr. Mason
Name:
Pythagorean Triple Worksheet
A formal definition: “A Pythagorean triple is a triple of positive integers a, b, and c such
that a right triangle exists with legs a & b and hypotenuse c.”
Given a & b, it is easy to find c:
1)
a = 11, b = 60, c =
Can you find two sides of a Pythagorean triple given just one side? Pythagoras and the
Babylonians had a method. In modern symbols: a Pythagorean triple will be generated
according to the following pattern: (2m, m 2  1, m 2  1)
2)
For m = 2, what is the triple?
3)
For m = 7, what is the triple?
A “Primitive” Pythagorean triple is one in which there is no common factor among the three
terms.
4)
Which ones of the three above are primitive Pythagorean triples?
5)
Given the three numbers of any Pythagorean triple, can you find three numbers that
will always divide into one or more of the sides?
Euclid, in Book X, Proposition 29, Lemma 1 (a lemma is a short theorem used to prove a
longer theorem) showed that (primitive) triples can be generated by (v 2  u 2 ,2uv, v 2  u 2 )
where v > u and u & v are relatively prime and u and v are of opposite parity (one is even
and one is odd).
6)
Show how that would work with 8 & 15.
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2/9/2016
Advanced Geometry
Mr. Mason
7)
Name:
Pick two numbers of your own to show another set of triples from Euclid’s formula.
Another method of generation of Pythagorean triples depends on Fibonacci numbers:
1, 1, 2, 3, 5, 8, …
Take a sequence of 4 Fibonacci numbers, Fn, Fn+1, Fn+2, & Fn+3. A Pythagorean triple can be
expressed by ( Fn  Fn 3 , 2  Fn 1  Fn  2 , ( Fn 1 ) 2  ( Fn  2 ) 2 )
8)
Pick a Fibonacci number greater than 21 and show how this expression generates a
Pythagorean triple.
A “Pythagorean Triangle” is a right triangle with integer sides (in other words, sides which
are a Pythagorean triple).
9)
Find the area of a half-dozen Pythagorean triangles:
10)
What do you notice about all those numbers?
11)
Besides being even, what other number divides evenly into all of them?
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Advanced Geometry
Mr. Mason
Name:
Here’s a right triangle with an incircle.
12)
Visualize the triangle as made up of
3 triangles whose bases are the sides of the
original triangle and the heights are all r.
Write an expression for the area of the
original triangle as a sum of three areas
using a, b, c, and r:
Area =
ab
:
2
13)
Solve your expression for r, and replace “Area” by
14)
What is the inradius of a 3-4-5 right triangle?
15)
Use Euclid’s expression for a, b, & , of a right triangle in the formula for inradius, and
simplify:
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Advanced Geometry
Mr. Mason
Name:
16)
Can the inradius of a Pythagorean triangle be anything other than an integer?
17)
There are only two Pythagorean triangles whose area and perimeter are the same
value. Find them.
18)
The number of primitive Pythagorean triangles with hypotenuse less than N is
approximately N/2 . How many primitive triples should there be with hypotenuse
less than 100?
19)
Count them on the list – how many are there?
20)
Find three consecutive numbers that are the hypotenuses of Pythagorean triangles.
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