Chord-Chord Relationships
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Introduction
    In this section of MATHguide, you will learn a variety of relationships within a circle that involve chords and arcs. Here are the topics within this page:




    Within this page, we are going to examine segments within a circle. Segments that have their endpoints on a circle are called chords. The figure below shows chord AB.

    The longest chord of a circle has a special name. It is called a diameter, which is shown below as chord CD.

    Notice how chord CD passes through the center of the circle.

    Learn more about other terminology related to circles in this MATHguide lesson.

    esson: Circle Terminology

    The sections that follow will address various properties that involve either partial lengths of chords or arcs and angles.


    Within this section, we will explore the relationships of lengths between intersecting chords of a single circle. Examine this diagram to view intersecting chords RS and TU.

Intersecting Chords

    The two chords intersect at point I. Consequently, each chord is fractured into two pieces. Chord RS is split into segments IR and IS and chord TU is split into segments IU and IT.

    The mathematical relationship that exists can be written like this.

Chord-Chord Length Equation

    Sometimes it is more advantageous to write the relationship using words, like this.

Chord-Chord Pieces Relationships

    To see how this relationship is proven, watch this video.

    ideo: Proof of Chord-Chord Product Rule

    Here is an example of how the relationship is applied.

Example 1: Solve for the x-value shown in this diagram.

Chord-Chord Pieces Problem

    The diagram above has two intersecting chords. The relationship informs us that we must calculate the product of the pieces of one chord and set it equal to the product of the pieces of the other chord, like so.

Initial Equation

    Finishing the relationship, we get this algebra.

Algebra

    ideo: Chord-Chord Partial Length Relationship
    uiz: Circles: Partial Chord Lengths


Example 2: Solve for the x-value shown in this diagram.

Chord-Chord Pieces Problem

    We will multiply the pieces of one chord and set the product equal to the product of the pieces of the other chord, like so.

Initial Equation

    To solve, we will begin by using the distributive property. Then, simple algebra thereafter.

Algebra

    ideo: Chord-Chord Partial Length Relationship
    uiz: Circles: Partial Chord Lengths


    When two chords intersect, there are certain arcs and angles that share an interesting relationship. Using the diagram below, we will share that relationship.

Chord Chord Arc Angle Problem

    The relationship that ties together angles and arcs is this:

Chord Chord Arc/Angle Formula

    Notice how the intersecting chords also intersect the circle. Arcs AB and CD are both within angle AEB and its vertical angle CED.

    Another way to think of this relationship can be written like so.

Chord Chord Arc Angle Problem

    We will now use this relationship to solve two problems.

Example 1: Solve for the x-value shown within this diagram.

Chord Chord Arc Angle Problem

    The arcs that are opposite the angle marked by the 'x' are 70 degrees and 30 degrees. According to the relationship, we will solve this equation for the x-value.

Chord Chord Arc Angle Algebra

    ideo: Chord-Chord Arcs and Angle Relationship
    uiz: Circles: Chords, Angles, and Arcs


Example 2: Solve for the x-value shown within this diagram.

Chord Chord Arc Angle Problem

    We will solve this problem by taking half the sum of the arcs and setting the result equal to the angle, like so.

Chord Chord Arc Angle Algebra

    To solve this algebra problem, we will make the right side a fraction. Then, we will solve the resulting proportion by cross multiplying, like so.

Chord Chord Arc Angle Algebra

    ideo: Chord-Chord Arcs and Angle Relationship
    uiz: Circles: Chords, Angles, and Arcs


    View the following instructional videos.

    ideo: Chord-Chord Partial Length Relationship
    ideo: Proof of Chord-Chord Product Rule
    ideo: Chord-Chord Arcs and Angle Relationship


    The following quizmaster will check for your understanding.

    uiz: Circles: Partial Chord Lengths
    uiz: Circles: Chords, Angles, and Arcs


    There are no activities at this time.


    Here are lessons that are related to chords.

    esson: Circle Terminology
    esson: Central Angles
    esson: Inscribed Angles
    esson: Secant-Secant Relationships