7.4
Arcs and Chords
An arc is a connected portion of a circle. Although two points of a line determine a unique segment of the line, the same is not true for two points of a circle. In the figure to the right, the points $A$ and $B$ break the circle into two parts. When we introduce a third point $C$ on the circle, there is no way to tell which of the three points is "between" the other two. If $C$ is a point of the circle on the large arc $\overparen{AB}$, you can start at $C$ and get to $A$ by passing through $B$. To distinguish the two arcs determined by $A$ and $B$, we draw the radii to these two points. When we do this, we can talk about the central angle $\angle APB$ that is formed. We call $A$, $B$, along with all the points of the circle that are in the interior of the central angle the minor arc $\overparen{AB}$. We call $A$, $B$, along with all the points of the circle that are in the exterior of the central angle the major arc $\overparen{AB}$. There is rarely any risk of confusion, but occasionally we will introduce another point and use a three-letter name such as $\overparen{ACB}$.
There is one time that this procedure doesn't work. If $A$ and $B$ are endpoints of a diameter, then $\angle APB$ is a straight angle and doesn't have an interior. Then we don't have a major arc and a minor arc. In that case, each of the two arcs $\overparen{AB}$is a semicircle.
The fact that we have minor arcs and major arcs, and that they are related to central angles of a circle, suggests that there be a way to measure arcs. In fact, there are two ways to measure arcs. Later on, we will consider the length of an arc; for now, we will measure an arc of a circle in terms of the measure of an angle.
- Definitions:
The degree measure of a minor arc
$\overparen{AB}$ of $\odot P$, denoted $m\overparen{AB}$, is the
measure of the central angle $\angle APB$.
The degree measure of a major arc of a circle is 360 minus the degree measure of the corresponding minor arc.
The degree measure of a semicircle is 180.
For now, we will normally omit the word "degree" when referring to the measure of an arc, since we have only one way to measure arcs.
We also introduce a postulate for arc measures. This is intimately tied to the Angle Addition Postulate (in fact, we could prove this postulate and have it as a theorem, but there are so many individual cases to consider that it would be a waste of time).
- Postulate:
The Arc Addition Postulate
If $B$ is a point on $\overparen{AC}$, then $m\overparen{ABC}=m\overparen{AB}+m\overparen{BC}.$
This postulate applies to major arcs and semicircles as well as to minor arcs. In the figure to the right, \begin{align} m\overparen{ABC}&=m\overparen{AB}+m\overparen{BC}\\ m\overparen{ADC}&=m\overparen{AD}+m\overparen{DC}\\ m\overparen{BC}+&m\overparen{CD}=m\overparen{BCD}=180 \end{align} |
We have the same sense for congruence of arcs that we have for other figures: two arcs should be congruent to each other only if one of them can be fit precisely on top of the other. That won't happen if the arcs occur in circles with different radii. This situation is pictured to the right, where both $\overparen{AB}$ and $\overparen{XY}$ have the same central angle $\angle P$. That is, $m\overparen{AB}=m\overparen{XY}$, but $\overparen{AB}\neq\overparen{XY}$. Congruence of arcs is tied to more than just the arc measures. We say that in the same circle or in congruent circles, congruent arcs are arcs that have the same measure. |
We now have a couple of very straightforward theorems that relate arcs to chords.
- Theorem 7.9:
In the same or congruent circles, congruent arcs have congruent chords.
We prove this theorem for the general case of two arcs in two congruent circles. The proof for two arcs in the same circle is essentially the same.
Given: $\odot P\cong\odot Q$
$\overparen{AB}\cong\overparen{XY}$
Prove: $\overline{AB}\cong\overline{XY}$
Proof:
\begin{array}{ l| l } Statement & Reasoning \\ \hline
1. \odot P\cong\odot Q\;;\;\overparen{AB}\cong\overparen{XY} & 1. \text{ Given} \\
2. \overline{PA}\cong\overline{QX}\;;\;\overline{PB}\cong\overline{QY} & 2. \text{ Def. of congruent circles}\\
3. m\overparen{AB}=m\overparen{XY} & 3. \text{ Def. of congruent arcs}\\
4. m\overparen{AB}=m\angle APB\;;\;m\overparen{XY}=m\angle XQY & 4.\text{ Def. of arc measure}\\
5. m\angle APB=m\angle XQY &5.\text{ Substitution}\\
6. \triangle APB\cong \triangle XQY & 6. \text{ SAS}\\
7. \overline{AB}\cong\overline{XY} & 7. \text{ CPCTC}
\end{array}
The converse of this theorem is also true.
- Theorem 7.10:
In the same or congruent circles, congruent chords have congruent arcs.
The proof is left as an exercise.
We finish with a definition and example.
- Definition:
$B$ is the midpoint of $\overparen{AC}$ if, and only if, $B$ is on
$\overparen{AC}$ and $\overparen{AB}\cong\overparen{BC}$.
Example: Given: $M$ is the
midpoint of $\overparen{AB}$ in $\odot O$. Prove: $\angle OAM \cong \angle OBM$ |
Proof: Since $M$ is the midpoint of $\overparen{AB}$, we know that $\overparen{AM}\cong\overparen{MB}$. Then $\overline{AM}\cong\overline{MB}$, since congruent arcs have congruent chords. The radii $\overline{OA}$, $\overline{OM}$, and $\overline{OB}$ are all congruent, so $\triangle OAM \cong \triangle OBM$ (by SSS). Hence $\angle OAM\cong\angle OBM$ (by CPCTC).