Leonel Contreras

2022-06-24

How is the Pythagorean Theorem related to the Equation of a Circle
My teacher wants me to figure out how the Pythagorean Theorem and the Equation of a Circle are related. I can't figure this out because I view them as being two completely different things. I understand what the equation for a circle is, but how does that relate to the Pythagorean Theorem?

Lilliana Burton

Expert

Draw a circle with radius r and center at the origin. In polar coordinates, $x=r\mathrm{sin}\theta ,y=r\mathrm{cos}\theta$. ${x}^{2}+{y}^{2}=1$ is thus true.
The equation for a circle is $\left(x-a{\right)}^{2}+\left(y-b{\right)}^{2}={r}^{2}$
You can use the following diagram:

Dale Tate

Expert

$\left(x-a{\right)}^{2}+\left(y-b{\right)}^{2}={r}^{2}$
Look at the equation of a circle with center (a,b) and radius r.
Consider the right triangle with sides A=x−a,B=y−b and C=r. You see that the equation of the circle is just the Pythagorean theorem.

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