Consider the following scenario. Sam is jogging along a circular-shaped track. Suddenly, he sees his friend, Max, on the other side of the jogging track and calls to greet him. Now, instead of covering the remaining distance along the jogging track, Sam walks in a straight line through the middle of the circle to meet up with Max. In mathematics, a straight line such as this is called a chord.
A chord can also be formed in any type of curve, such as ellipses. Here, we will particularly discuss the properties of chords in circles.
Properties of a chord in a circle
Before discussing the properties of a chord in a circle, let's quickly consider the definition of chord. A chord is a line segment that passes from any two points in a circle. A chord can be drawn with its two endpoints anywhere on the circle.
A chord of a circle is a line segment that has its endpoints on a circle.
When a chord passes through the center of the circle, we refer to it as the circle's diameter. A diameter divides a circle into two semicircles, whereas any other chord splits a circle into a major arc and a minor arc.
Fig. 1. Chords of the circle.
The basic properties of a chord are as follows:
The chord divides a circle into two segments: major segment and minor segment.
A perpendicular line bisects the chord if drawn from the center of the circle to the chord.
Two equal chords have two equal angles subtended at the center of the circle.
Chords with equal length are equidistant from the center of the circle.
Now let's dive deeper and understand the properties of chords more clearly.
Properties of chords: Perpendicular bisectors
Suppose that you have a chord \(AB\) on a circle having a center \(O\), which the figure below illustrates. If we draw a line from the center \(O\) to the point \(P\) on the chord \(AB\), \(OP\) is perpendicular to \(AB\), and \(OP\) bisects \(AB\). In other words, \(OP\) is the perpendicular bisector of \(AB\) such that \(AP\) and \(PB\) are congruent. Hence,
\[\text{if} \; \overline{OP}\perp \overline{AB} \text{, then} \; \overline{AP}\cong \overline{PB}\]
Fig. 2. Chord with perpendicular bisector.
Determining the diameter by a perpendicular bisector
Have a look at the below figure. \(CD\) is the chord that acts as a perpendicular bisector to the chord \(AB\). For this scenario,
\[\text{if} \; \overline{CD}\perp \overline{AB} \text{, then} \; \overline{CD} \; \text{is a diameter of the circle}\]
Fig. 3. Chord with perpendicular bisector.
Properties of congruent chords
If two chords are equidistant from the circle's center, we know that they must be congruent. This property of chords is depicted in the figure below: chords \(AB\) and \(DE\) are equidistant on the circle. Also note that \(CF\) and \(CG\) are equal in length (congruent). The equal length of these line segments \(CF\) and \(CG\) help us confirm that the two chords \(AB\) and \(DE\) are of equal distance from the circle's center.
\[\text{if} \; \overline{CF}\cong \overline{CG} \text{, then} \; \overline{AB}\cong \overline{DE}\]
Fig. 4. Chords with equal lines from center.
Now, let's say we are given the information that \(CF\) is perpendicular to \(AB\), and \(CG\) is perpendicular to \(DE\). In this case, we can use the property of perpendicular bisectors to make the following conclusions about the chords: \[\text{if} \; \overline{CF}\perp \overline{AB} \; \text{and} \; \overline{CG}\perp \overline{DE} \; \text{and} \; \overline{CF} = \overline{CG} \; \text{, then} \; \overline{AF} = \overline{FB} = \overline{DG} = \overline{GE}\]
Fig. 5. Perpendicular chords with equal sides.
Properties of intersecting chords
The intersecting chords theorem states that when chords in a circle intersect, the products of their segments' lengths are equal. Have a look at the figure below, where two chords \(RS\) and \(PQ\) intersect at point \(A\), with \(O\) as the center of the circle. So, we can write the chord theorem as:
\[(\overline{SA}) \cdot (\overline{AR}) = (\overline{PA}) \cdot (\overline{AQ})\]
Fig. 6. Intersecting chords.
Properties of chords: Subtended angles
The next property of chords we'll discuss deals with subtended angles. First, let's clarify the meaning of a subtended angle. When the two endpoints of a chord are joined (using line segments) to form an angle located at a point outside of that chord, that angle is considered the subtended angle.
For example, suppose \(AB\) is a chord and \(C\) is a point outside the chord in a circle. Then \(\angle ACB\) is the subtended angle.
Fig. 7. Chord with subtended angle.
Now, let's consider the next chords property by having a look at the figure below: in this figure, two equal chords subtend angles at the center of a circle. In this case, according to the properties of chords, both subtended angles are equal.
\[\Rightarrow \angle AOB =\angle DOC \]
Fig. 8. Two equal chords subtend equal angles at the center.
Properties of chords: Calculation of length with formulas
In certain circumstances, we are able to calculate the length of a chord using formulas, including:
- When the chord's subtended angle at the center of the circle is provided.
- When the radius and distance from the chord to the center is given.
These circumstances are shown in the figure below. Suppose for the chord \(CB\) on the circle with center \(A\), \(r\) is the radius, \(d\) is the distance from the chord to the center, and \(\theta\) is the subtended angle.
Fig. 9. Chord length with radius or subtended angle.
The length of chord \(CB\) shown in the figure can be calculated by use of the following formulas:
When the subtended angle is given, then:
\[\text{Chord}=2 \times r \times \sin \left ( \frac{\theta}{2} \right )\]
If the radius and distance from the center to the chord is given, then:
\[\text{Chord}=2 \times \sqrt{r^{2}-d^{2}}\]
Properties of chords examples
Let's exercise our knowledge of the properties of chords with some example problems.
Have a look at the circle below with chords \(AB\) and \(DE\). \(C\) is the center of the circle, with \(CF\) and \(CG\) bisecting \(AB\) and \(DE\), respectively. For the circle below:
Part A:
\begin{align}&\overline{AB}=\overline{DE} \\&\overline{CF}=3x+16 \\&\overline{CG}=6x+10\end{align}
Calculate \(\overline{FG}\).
Part B:
\begin{align}&\overline{CG} \perp \overline{DE} \\&\overline{DG}=8x-17 \\&\overline{DE}=4x+14\end{align}
Calculate \(\overline{DE}\).
Fig. 10. Two chords with bisecting lines from center.
Solution:
Part A:
As the chords \(AB\) and \(DE\) are equal, we can conclude that \(CF\) and \(CG\) must be equal as well. This is because two chords are equal to each other if they are equidistant from the center of the circle.
Therefore,
\begin{align}&\overline{CF}=\overline{CG} \\&3x+16=6x+10 \\&3x=6 \\&x=2\end{align}
Hence, we get \(\overline{CF}=\overline{CG}=22\).
And,
\begin{align}&\overline{FG}=\overline{FC}+\overline{CG} \\&\overline{FG}=22+22 \\&\overline{FG}=44\end{align}
Hence, \(\overline{FG}=44\).
Part B:
As CG and DE are perpendicular, DG and GE are equal.
Hence,
\begin{align}&\overline{DE}=\overline{DG}+\overline{GE} \\&\overline{DE}=2(\overline{DG}) \\&4x+14=2(8x-17) \\&x=4\end{align}So, the chord \(DE\) is equal to:
\begin{align}\overline{DE}&=4x+14 \\&=4(4)+14 \\&=30\end{align}
Hence, \(\overline{DE}=30\).
Calculate the distance from the center of the circle to the chord if the chord is \(16\, \text{cm}\) and the diameter is \(20\, \text{cm}\).
Solution:
To visualize this better, have a look at the figure below.
Fig. 11. Chords with the distance from the center.
For the circle above:
\begin{align}&\overline{DE}=20\, \text{cm} \\&\overline{AB}=16\, \text{cm} \\\end{align}
As \(DE\) is the diameter, \(CE\) and \(CD\) are \(10\, \text{cm}\). The dotted line \(AC\) also happens to be the radius of the circle. Hence,
\[\overline{AC}=10\, \text{cm}\]
The distance \(CF\) is the perpendicular bisector that we have to calculate. As \(CF\) is the perpendicular bisector, it will cut the chord \(AB\) into two halves, and hence:
\[\overline{BF}=\overline{FA}=8\text{ cm}\]
To calculate \(CF\), we will use the Pythagorean theorem which results in:
\begin{align}&AC^{2}=AF^{2}+FC^{2} \\&10^{2}=8^{2}+FC^{2} \\&\overline{FC}=6\text{ cm}\end{align}
So, the distance from the center of the circle to the chord is \(6\text{ cm}\).
Properties of chords - Key takeaways
- A chord is a line segment that has its endpoints on a circle.
- A chord divides the circle into a major arc and a minor arc.
- Two equal chords subtend equal angles at the center.
- A chord is a diameter if it is a perpendicular bisector of another chord.
- If two chords are equidistant from the circle's center, then they are congruent.
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