Circles arise naturally in geometry (they are an immediate consequence of the notion of distance), and we use them extensively in geometric constructions (after all, a compass is essentially a “circle-drawing tool”). Today, we look at some geometrical facts about circles, and then we use the coordinate plane to make the connection between the geometry and algebra (equations) of circles.

Circle

Definition. Given a point $P$ and a positive real number $r$, the circle with center $P$ and radius $r$ consists of all points in the plane of distance $r$ from $P$.

Circles and Line Segments

A diameter of a circle is any line segment with both endpoints on the circle which passes through the center $P$. Note: Sometimes we use the word “diameter” to mean the length of such a segment.

A radius of a circle is any line segment with one endpoint at the center $P$ and the other endpoint on the circle. NOTE: Once again, we sometimes use the word “radius” to mean the length of such a segment.

Chords and Arcs

A chord of a circle is any line segment with both endpoints on the circle.

Arc. If $P$ and $Q$ are distinct points on a circle, then we can imagine the chord $\overline{PQ}$ dividing the circle into two parts, called arcs. When we use the notation $\stackrel{\frown}{PQ}$ it is usually clear from context which of these two arcs we mean – but sometimes, an additional letter is provided to help the reader.

Example: In the image below $\stackrel{\frown}{PAQ}$ refers to the upper arc.

The chord PAQ

Drawing angles in circles

Theorem. If $\overline{PQ}$ is a diameter of a circle and $A$ is any other point on the circle, then the angle $\angle{PAQ}$ is a right angle.

The angle PAQ inside the circle with diameter PQ

What if we use a different chord (not a diameter)?

Theorem. If $\overline{PQ}$ is a chord of a circle and $A$ is any other point on the circle, then the measure of $\angle{PAQ}$ does not depend on the choice of $A$ – we call this the angle subtended by $\stackrel{\frown}{PQ}$.

The angle PAQ inside the circle with chord PQ

This angle is $90^\circ$ if and only if the chord $\overline{PQ}$ is a diameter.

From Geometry to Algebra

When we start using x and y axes, we suddenly we gain the ability to refer to points by pairs of numbers, or coordinates. This allows us to use the tools of algebra to talk about the shapes from geometry. We call this topic analytic geometry.

Distance

Definition. Given two points $(x_1,y_1)$ and $(x_2,y_2)$, the distance d between them is given by $d=\sqrt{(x_2-x_1)^2 + (y_2-y_1)^2}$.

Question: Suppose we have a circle of radius r=5 centered at (4,6). If (x,y) is a point on the circle, what is the distance from (4,6) to (x,y)? Express using the definition of distance.

GROUPS: Find the equation for the circle with center $(-2,\frac{3}{4})$ and radius $5.3$. Is the point $(1.1, 4)$ on the circle? Give 3 points on the circle.

GROUPS: Find the center and radius of the circle given by the equation $x^2 + y^2 -6x + 14y + 54 = 0$.

GROUPS: Find the center and radius of the circle given by the equation $x^2 -\frac{x}{3} + y^2 + \frac{50 y}{3} + \frac{737}{36} = 0$