Circles
Definitions
A circle is defined as the set of all points of some fixed distance from a central point. There are other examples of definitions like this. Any such point in a definition like this is called a "locus" and the plural form is "loci".
For example, the set of all points equally distant from any two distinct points is a line. Explore this idea in the following way.
Take any two arbitrary points like A and B below. Suppose line g is the perpendicular bisector of (the segment is not depicted here). Suppose C is on line and on line g. Show that the distance AC = CB. Now pick an arbitrary other point on line g like point D, and prove that AD = DB. This is a proof that the line is the set of all points equally distant from A and B.
So
The locus of all points equally distant to two points A and B is a line, and is their perpendicular bisector.Similarly,
The locus of all points with a distance of r from a point A is a circle. We call r the radius and A the center.
Problem 1.
What is the locus of all points equally distant from two parallel lines?
Problem 2.
What is the locus of all points equally distant from two intersecting lines?
Problem 3.
Try to guess what shape is made by the locus of all points equally distant from a point and a line.
More Definitions
There are two ways to measure a circle, we can either measure its circumference or its area. The circumference is like the perimeter for a polygon, it's the length around the outside. The formula for the circumference of a circle is where r is the radius of the circle. is a number, and we don't really need to know its value because we will not use the value any time soon. It may be worth knowing that, if d is the diameter of the circle (twice as big as the radius) then the equation is the definition of .
The formula for the area of a circle is .
Problem 4.
a) If the radius of a circle is 10 find its diameter, circumference, and area.
b) If the diameter of a circle is 10, find its radius, circumference, and area.
c) If the circumference of a circle is 10, find everything else.
d) If the area of a circle is 10, find everything else.
e) If the area of a circle is find everything else.
More Definitions
We define a few notions of "angles" and "arcs". First there is the central angle, depicted below.
The central angle is the angle made at the center, colored green here. In this case it .
If c is a circle centered at point O with points A and B on it, then is called a central angle.The red curve connecting B and C is called the arc between B and C. Notice that an arc exists starting from B and rotating clockwise to C, and this arc makes up the rest of the circle. We define "the arc between B and C" to be the shortest arc connecting the points--hence the red arc in this picture is the arc between them. We use the symbol to denote the arc between A and B.
If c is a circle with points A and B, the portion of the circle that forms the shortest curve starting at A and ending at B is the arc between A and B.
Problem 5.
Although the central angle is different from the arc, there is clearly a relationship. Let's try to find it. Whenever we're trying to figure something out we try to pick easy versions of the problem and build up to harder ones. So let's start with a 90-degree angle. Suppose a circle has center at O and points A and B on it, and its radius is 36. If . Find then length of 
.
What would it be if the measure of the angle were 180? 270? 360? 45? 10?
Suppose the measure of the arc is x and we don't know what x is. What is the length of the arc in terms of x?
.
What would it be if the measure of the angle were 180? 270? 360? 45? 10?
Suppose the measure of the arc is x and we don't know what x is. What is the length of the arc in terms of x?Problem 6.
We also define a sector of a circle as being that filled-in region within an angle. I have depicted it below.
If the radius of this circle is 1 and , find the area of sector ABC.
More Definitions
The length of the arc is not the only way that we measure the arc. Another way to measure it is to use degrees. Just like the central angle can make a 360-degree rotation, which completes the circle, so too the arc around the edge of the circle can make a 360-degree rotation and complete the circle. Thus the following picture shows. Whatever is the measure of the central angle, that will also be the measure of the arc. Thus in the picture above .
Finally we'll define an "inscribed angle". The picture below shows one.
The inscribed arc is . We call the arc from B to D that does not contain C the "intercepted arc" because it kind of looks like the angle hits the arc.
This is something to really study. How does the inscribed angle relate to the intercepted arc?
Let's start with a warm-up problem.
Problem 6.
In the figure below suppose that the line bisects . If then find .
Problem 7.
Let's do another variation on this, this time where we know that half of the inscribed angle is a diameter, like this.
Here the angle has diameter as one of its sides. I've labeled . In this case it should be very easy (if you the right things) to find .
Problem 8.
Next we'll try something harder. Suppose point D is located anywhere on the circle, and let's see if we can find a relationship between and .
We want to know what the relationship is between and . We've already seen (hopefully) that when point D is located directly behind A then the angle is half as big. But what about now, where D could be anywhere on the circle? This is a pretty hard problem. However, I'll set you up with a lot of hints. I will draw and label a bunch of angles on this picture. Try to find them all.
By including the diameter you now can use the fact that just like we showed in the previous problem. Using that, together with the fact that you should be able to show that where the ... is something that only has z in it.