Pythagoras theorem is a^2 + b^2 = c^2 and a circle has an equation x^2 + y^2 = a^2 .Is there a relation between a right angle triangle and a circle?

Question

I was just curious about the fact that whether such a relation exists when I came across the equation of a circle.(I maybe absolutely wrong) .

Answer 1

Yes they're related! A circle is the locus of a point, which is always equidistant from the center.

Now $P$ is a point on the circle of radius $r$ with co-ordinates $(x,y)$.

By Pythagoras theorem

$r^2=x^2+y^2$

Which is the equation of the circle!

Answer 2

If we have a segment $AB$ and $O$ is the midpoint, a circle is formed by all possible locations for the third vertex of a right triangle that has $AB$ as the hypotenuse and $O$ will be the center of the circle.

Answer 3

This relationship can be summed up as:

For any right triangle there exists only one semi-circle whose diameter is equal hypotenuse of the right triangle.

Conversely: For any semi circle there exists infinitely many right triangles whose hypotenuse is equal to the diameter of the semi-circle