Pythagorean theorem vs an equation of a circle?

Question

Today I told my teacher that the equation of a circle looks like to the Pythagorean theorem to me, but he said that I'm wrong and to re think it.

Why $(x-h)^2 + (y-k)^2 = r^2$ is not a PT, it looks just like PT -- we square two numbers, add them and we get another number squared.

Where $(h,k)$ is the center and $r$ is the radius of a circle.

Can someone explain in more details?

Answer 1

Suppose the center of the circle is (0,0) and $(x,y)$ is a point of the circle of radius $r$, $x^2+y^2=r^2$

The points $A=(0,0), B=(y,0), C=(x,y)$ define a rectangle triangle.

Answer 2

I can NOT believe you teacher said that! Take a circle with radius r centered at (h,k). Pick an arbitrary (x,y) on the circle. Drop a vertical line down from (x,y) to (x, k). Slide a horizontal line from (h,k) to (x,k). These three points (x,y) to (x,k) to (h,k) back to (x,y) form a triangle. Not just a triangle a right triangle.

So the triangle has base side A = (h,k) to (x,k). It has length (x - h).

The triangle has height side B = (x,k) to (x, y). It has length (y - k).

The triangle has hypothenuse C = (h,k) to (x,y). It has lenght r.

The pythagorean theorem says $A^2 + B^2 = C^2$ or $(x-h)^2 + (y - k)^2 = r^2$. This gives us a formula for any point of the circle that relates the x value to its y value.

It is base entirely upon the Pythagorean Theorem.

Answer 3

When you write the equation of a circle, you start from a definition as:

The locus of points $P$ that have the same distance $r$ from a fixed point $C$ (the center).

Then you write the equation as $\overline PO=r$ using the standard definition of (Pythagorean) distance for points represented in an orthogonal reference system: $$ \overline PO=\sqrt{(x_P-x_O)^2+(y_P-y_O)^2}=r $$ and squaring you find the equation $$ (x_P-x_O)^2+(y_P-y_O)^2=r^2 $$ Clearly, this seems the classical pythagorean formula, and a figure confirm that there is a rectangular triangle with $r$ as hypotenuse. But note that this is a consequence of many other assumptions: the definition of the circle, the use of a particular distance, an the use of an orthogonal reference system. And, finally, your equation refers to an infinite set of points $P$ not to a single triangle.

So, in my opinion, your teacher is right in inviting you to better think to the question and to the difference between a theorem and a consequence of this theorem.

In a different coordinate system, the equation of a circle can be very different, but the PT is always the same if the metric (the distance) is Pythagorean.