In the below activity grab point P and drag it around the the graph. Notice the trace it creates.
We see that when moving around the point P that we create a circle. [br][br]A circle is the locus of points in a plane equidistant from a given point in the same plane. The given point is called the center and the given distance is the radius. [br][br]See the activity below and observe that when point P moves around the circle, the equation shown doesn't change.
So, we have a right triangle. We know side a is the y component of point P and side b is the x component. [br][br]The Pythagorean Theorem is [math]a^2+b^2=c^2[/math] where [math]c=\sqrt{a^2+b^2}[/math]. This is used to find the length of the hypotenuse, h. The equation of a circle is [math]x^2+y^2=r^2[/math], look familiar?? [br]The sides of the right triangle and the hypotenuse are used in building the circle and thus its equation.
Since P lies on the circle, what must be true about its coordinates? Pick a point and verify that it satisfies your answer.
The coordinates of point P must satisfy the equation of the circle. For example, if the coordinate is substituted into the Pythagorean Theorem or the equation of circle then x^2 + y^2 must be equal to the radius squared. [br][br]For example, (3, 4) will give you 25 and hence the circle will have a radius of 5.
In the activity below, change the radius of circle A by dragging the slider back and forth. [br][br]Notice what changes in the equation of the circle.
So, we see that by changing the radius of the circle, the only thing in the equation that changes is what the equation equals. Much like in the pythagorean theorem, when c changes, the hypotenuse changes, so when the radius changes, the circle gets bigger/smaller. From this, we can conclude that the hypotenuse of the right triangle = the radius of a circle. [br][br]Now we can relate the Pythagorean Theorem to the Equation of a circle:[br][br][math]a^2+b^2=c^2\Longleftrightarrow x^2+y^2=r^2[/math]