Circles
Acircleis the set of all points in a plane at a given distance (called theradius) from a given point (called the center.)
A line segment connecting two points on the circle and going through the center is called adiameterof the circle.
Clearly, if代表一个直径的长度ndrepresents the length of a radius, then.
Thecircumference of a circle is the distance around the outside. For any circle, this length is related to the radiusby the equation
where(pronounced "pi") is anirrationalconstant approximately equal to.
Theareaof a circle is given the formula
.
It can be shown that any two circles in the plane aresimilar, as follows.
Proof that any two circles are similar
Suppose we have two circles, circlecentered atwith radiusand circlecentered atwith radius.
First, we translate circle Aunits to the right andunits up, so that it is now centered. (Note thatand/or可能是负的,在这种情况下,我们实际上是史fting the circle left and/or down.)
Then, we perform a dilation, centered at, by a scale factor of. This results in a circle centered atwith a radius of.
That is, we have transformed circleinto circle, using nothing but translation and dilation. Therefore, the two figures are similar.