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Maths Exercices
Converse of the Pythagorean Theorem

The Pythagorean Theorem says:

Pythagorean Theorem
In a right triangle, with legs a and b, and hypotenuse c, a2+b2=c2

But what about the converse of the Pythagorean Theorem?

Converse of the Pythagorean Theorem
In a triangle with sides a, b and c, if a2+b2=c2 then it is a right triangle

Are these right triangles?

Solution:
We can use the Converse of the Pythagorean Theorem to find if the triangles are right triangles.
If the equation a2+b2=c2 is true, then we will have a right triangle.

For the first triangle:

 a2+b2=c2
122+352=362

The first triangle is not a right triangle.


For the second triangle:

 a2+b2=c2
 152+362=392
1521=1521
The second triangle is a right triangle

To concrete examples of the converse are the Pythagorean triples:

 

A Pythagorean triple consists of three positive integers a, b, and c, such that a2 + b2 = c2. A right triangle whose sides form a Pythagorean triple is called a Pythagorean triangle.

Such a triple is commonly written (a, b, c), and well-known examples are:

(3, 4, 5), (5,12,13), (8,15,17), (7,24,25).

This is the right triangle formed by the Pythagorean triple (3,4,5)
This is the right triangle formed by the Pythagorean triple (5,12,13)
This is the right triangle formed by the Pythagorean triple (8,15,17)
This is the right triangle formed by the Pythagorean triple (7,24,25)

If (a, b, c) is a Pythagorean triple, then so is (ka, kb, kc) for any positive integer k.

A primitive Pythagorean triple (PPT) is one in which a, b and c are pairwise coprime.

There is a great formula that will generate an infinite number of Pythagoren Triples that are integers (not fractions).

Formula for generating Pythagorean Triples
Choose any two positive integers - call them s and t. Let s be larger than t. Then the following three numbers are a Pythagorean Triple:

{2st, s2-t2, s2+t2}

For instance, if s=5 and t =1, then 2st=10, s2-t2=25-1=24 and s2+t2=25+1=26.

You get the Pythagorean Triple (10,24,26)

Substituting those numbers into the Pythagorean Theorem:

102+242=262
100+576=676
676=676