Properties of Chords

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.

Properties of Chords Properties of Chords

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    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.

    Properties of chords, chords, StudySmarterFig. 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,

    \[\text{if} \; \overline{OP}\perp \overline{AB} \text{, then} \; \overline{AP}\cong \overline{PB}\]

    Properties of chords, chords with perpendicular bisector, StudySmarterFig. 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,

    \[\text{if} \; \overline{CD}\perp \overline{AB} \text{, then} \; \overline{CD} \; \text{is a diameter of the circle}\]

    Properties of chords, chords with perpendicular bisector, StudySmarterFig. 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.

    \[\text{if} \; \overline{CF}\cong \overline{CG} \text{, then} \; \overline{AB}\cong \overline{DE}\]

    Properties of chords, chords with equal sides, StudySmarterFig. 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: \[\text{if} \; \overline{CF}\perp \overline{AB} \; \text{and} \; \overline{CG}\perp \overline{DE} \; \text{and} \; \overline{CF} = \overline{CG} \; \text{, then} \; \overline{AF} = \overline{FB} = \overline{DG} = \overline{GE}\]

    Properties of chords, chords with perpendicular bisector, StudySmarterFig. 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:

    \[(\overline{SA}) \cdot (\overline{AR}) = (\overline{PA}) \cdot (\overline{AQ})\]

    Properties of chord, intersecting chords, StudySmarterFig. 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.

    Properties of chords, chords with subtended angle, StudySmarterFig. 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.

    \[\Rightarrow \angle AOB =\angle DOC \]

    Properties of chords, equal angles of circle with equal chords, StudySmarterFig. 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.

    Properties of chords, chords, StudySmarterFig. 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 \(\overline{DE}\).

    Properties of chord, chords with bisecting lines, StudySmarterFig. 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.

    Properties of chords, chords, StudySmarterFig. 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.
    Frequently Asked Questions about Properties of Chords

    How do you solve properties of chords? 

    You can solve problems using the properties of chords by applying the intersecting or congruence properties.

    What are the chord properties of a circle?

    • A perpendicular bisector cuts a chord into two equal lengths. 
    • If two chords intersect such that the first chord is the perpendicular bisector, then the first chord is a diameter. 
    • When chords in a circle intersect, the products of their segments' lengths are equal. 
    • If two chords are equidistant from the circle's center, we know that they must be congruent. 

    What are the properties of intersecting chords?

    The intersecting chords theorem states that when chords in a circle intersect, the products of their segments' lengths are equal.

    What are the properties of a congruent chord?

    Two chords are congruent if they are equidistant from the center of the circle. 

    Test your knowledge with multiple choice flashcards

    A line segment that has its endpoints on a circle is called?

    A chord passing through the center of the circle is called?

    Does a chord, apart from a diameter, any other chord splits a circle into a major arc and a minor arc?

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