What is the mathematical significance of the numbers 25, 16 and 9? How
does a triangle fit into that? Please explain the significance.

Hi,

Suzyq-ga below is correct.  The Pythagorean theorem states that in a
right triangle (a triangle with a right angle) the square (a number
multiplied by itself) of each side that is part of the right angle is
equal to the square of the hypotenuse (the side opposite the right
angle).  If you don't understand all those terms, you can follow the
links below for pages that show diagrams.

Thus, to take the example that this question poses, we have a triangle
that has the sides of 3 units, 4 units and 5 units.  According to the
theorem, if this is a right angle then 3 squared plus 4 squared should
equals 5 squared, which it does, since 3^2=9, 4^2=16 and 5^2=25, and
9+16=25.

This particular right triangle is unusual, since 3, 4 and 5 are
consecutive integers.  They are the only consecutive integers that
form a right triangle.  It is also the smallest right triangle where
all three sides can be represented by integers.

If you want to know more about the Pythagorean theorem, see the
following links.  All of them use the above example:

The Pythagorean Theorem
http://www.themathpage.com/aTrig/pythagorean-theorem.htm

The Pythagorean Theorem
http://www.geocities.com/thesciencefiles/pythagoras/pythagoreantheorem.html

Pythagorean Triples
http://www.jimloy.com/number/triples.htm

I hope this helps your understanding.

Sincerely,

mvguy





Google search term: pythagorean theorem 3 4 5 9 16 25
://www.google.com/search?q=pythagorean+theorem+3+4+5+9+16+25

Provided excellent resources and a well thought out answer.

This sounds like an application of Pythagoras' Theorem; the length of
the hypotenuse of a right-angled triangle is the square root of the
sum of the two remaining sides squared.

So, 3^2 + 4^2 = 5^2
      9 + 16  = 25

Thanks. :)