What if you were asked to geometrically consider the ancient astronomical clock in Prague, pictured below? It has a large background circle that tells the local time and the “ancient time” and then the smaller circle rotates around on the orange line to show the current astrological sign. The yellow point is the center of the larger clock. How does the orange line relate to the small and larger circle? How does the hand with the moon on it (black hand with the circle) relate to both circles? Are the circles concentric or tangent?

### Parts of Circles

A **circle** is the set of all points in the plane that are the same distance away from a specific point, called the **center**. The center of the circle below is point \begin{align*}A\end{align*}. We call this circle “circle \begin{align*}A\end{align*},” and it is labeled \begin{align*}\bigodot A\end{align*}.

##### Important Circle Parts

**Radius:** The distance from the center of the circle to its outer rim.

**Chord:** A line segment whose endpoints are on a circle.

**Diameter:** A chord that passes through the center of the circle. The length of a diameter is two times the length of a radius.

**Secant:** A line that intersects a circle in two points.

**Tangent:** A line that intersects a circle in exactly one point.

**Point of Tangency:** The point where a tangent line touches the circle.

The tangent ray \begin{align*}\overrightarrow{TP}\end{align*} and tangent segment \begin{align*}\overline{TP}\end{align*} are also called tangents.

**Tangent Circles:** Two or more circles that intersect at one point.

Two circles can be tangent to each other in two different ways, either *internally* tangent or *externally* tangent.

If the circles are not tangent, they can share a tangent line, called a *common* tangent. Common tangents can be internally tangent and externally tangent too. Notice that the common internal tangent passes through the space between the two circles. Common external tangents stay on the top or bottom of both circles.

**Concentric Circles:** Two or more circles that have the same center, but different radii.

**Congruent Circles:** Two or more circles with the same radius, but different centers.

*Watch the first half of this video.*

#### Identifying Parts of Circles

Find the parts of \begin{align*}\bigodot A\end{align*} that best fit each description.

a) A radius

\begin{align*}\overline{HA}\end{align*} or \begin{align*}\overline{AF}\end{align*}

b) A chord

\begin{align*}\overline{CD}, \ \overline{HF}\end{align*}, or \begin{align*}\overline {DG}\end{align*}

c) A tangent line

\begin{align*}\overleftrightarrow{BJ}\end{align*}

d) A point of tangency

*Point H *

e) A diameter

\begin{align*}\overline{HF}\end{align*}

f) A secant

\begin{align*}\overleftrightarrow{BD}\end{align*}

#### Drawing Intersecting Cricles

Draw an example of how two circles can intersect with no, one and two points of intersection. You will make three separate drawings.

#### Determining if Circles are Congruent

Determine if any of the following circles are congruent.

From each center, count the units to the outer rim of the circle. It is easiest to count vertically or horizontally. Doing this, we have:

\begin{align*}\text{Radius of} \ \bigodot A & = 3 \ units\\ \text{Radius of} \ \bigodot B & = 4 \ units\\ \text{Radius of} \ \bigodot C & = 3 \ units\end{align*}

From these measurements, we see that \begin{align*}\bigodot A \cong \bigodot C\end{align*}.

Notice the ** circles are congruent**. The

**are**

*lengths of the radii***.**

*equal*Refer to the photograph at the beginning of this section. The orange line (which is normally black, but outlined for the purpose of this exercise) is a diameter of the smaller circle. Since this line passes through the center of the larger circle (yellow point, also outlined), it is part of one of its diameters. The “moon” hand is a diameter of the larger circle, but a secant of the smaller circle. The circles are not concentric because they do not have the same center and are not tangent because the sides of the circles do not touch.

### Examples

#### Example 1

If the diameter of a circle is 10 inches, how long is the radius?

The radius is always half the length of the diameter, so it is 5 inches.

#### Example 2

Is it possible to have a line that intersects a circle three times? If so, draw one. If not, explain.

It is not possible. By definition, all lines are *straight*. The maximum number of times a line can intersect a circle is twice

#### Example 3

Are all circles similar?

Yes. All circles are the same shape, but not necessarily the same size, so they are similar.

### Review

Determine which term best describes each of the following parts of \begin{align*}\bigodot P\end{align*}.

- \begin{align*}\overline{KG}\end{align*}
- \begin{align*}\overleftrightarrow{FH}\end{align*}
- \begin{align*}\overline{KH}\end{align*}
- \begin{align*}E\end{align*}
- \begin{align*}\overleftrightarrow{BK}\end{align*}
- \begin{align*}\overleftrightarrow{CF}\end{align*}
- \begin{align*}A\end{align*}
- \begin{align*}\overline{JG}\end{align*}
- \begin{align*}\overline{HG}\end{align*}
- What is the longest chord in any circle?

Use the graph below to answer the following questions.

- Find the radius of each circle.
- Are any circles congruent? How do you know?
- How are \begin{align*}\bigodot C\end{align*} and \begin{align*}\bigodot E\end{align*} related?
- Find the equation of \begin{align*}\overleftrightarrow{CE}\end{align*}.
- Find the length of \begin{align*}\overline{CE}\end{align*}.

### Review (Answers)

To view the Review answers, open this PDF file and look for section 9.1.