Topics
9. Circles
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United States of AmericaFL
Grade 912

9.03 Chords

Lesson
Practice
Lesson

Concept summary

Between any two points on a circle, we can define the following line segment:

Chord

A line segment that connects two points on the arc of a circle. For example, \overline{AB} is a chord.

Points A and B on a circle. A segment is drawn from A to B.

The following theorems each relate to properties of one or more chords in a circle:

Congruent corresponding chords thoerem

Two chords in a circle, or in congruent circles, are congruent if and only if the corresponding arcs are congruent.

Circle C with chords A B and X Y. Minor arcs A B and X Y are congruent. Chords A B and X Y are congruent.
Corollary to the corresponding chords theorem

Two chords in a circle, or in congruent circles, are congruent if and only if the corresponding central angles are congruent.

Circle C with chords A B and X Y. Chords A B and X Y are congruent. Central angles A C B and X C Y are congruent.
Equidistant chords theorem

In the same circle, or congruent circles, two chords are congruent if and only if they are equidistant from the center.

Circle C with chords A B and X Y. Chords A B and X Y are congruent. The perpendicular distance from A B to C, and from X Y to C are congruent.
Perpendicular chord bisector theorem

A diameter bisects a chord and its corresponding arc if and only if the diameter is perpendicular to the chord.

Circle C with diameter A B and chord X Y. X Y is perpendicular to A B. A B divides chord X Y into 2 congruent segments and minor arc X Y into 2 congruent arcs.

Worked examples

Example 1

Given NP=4x and LM=6x-9, find LM.

Circle Q is drawn with chord P N and chord L M opposite each other. Segment Q S is drawn perpendicular to chord P N. Segment Q R is drawn perpendicular to chord L M. Segment Q S and segment Q R are marked congruent.

Approach

We have been given that the two chords are equidisant from the center of the circle. We can use the equidisant chords theorem to set up and equation so we can solve for x and then evaluate LM=6x-9.

Solution

\displaystyle LM\displaystyle =\displaystyle NPEquidisant chords theorem and definition of congruence
\displaystyle 6x-9\displaystyle =\displaystyle 4xSubstitution
\displaystyle 2x\displaystyle =\displaystyle 9Add 9 and subtract 4x from both sides
\displaystyle x\displaystyle =\displaystyle \frac{9}{2}Divide both sides by 2

Example 2

In the diagram below, we are given that \angle BAC \cong \angle EAD, CB = 2x + 15, and ED = 5x -3.

Solve for x.

Two triangles, A B C and A E D, inside circle A. Segments C B and E D are chords of the circle. Angles E A D and B A C are congruent.

Approach

We know that \angle BAC \cong \angle EAD, and these are the central angles of the chords CB and DE. So by corollary to the corresponding chords theorem, we know that the chords are also congruent. We can now write an equation relating the two lengths of the chords and solve for x.

Solution

\displaystyle 2x + 15\displaystyle =\displaystyle 5x-3Equating chord lengths
\displaystyle 2x+18\displaystyle =\displaystyle 5xAdd 3 to both sides
\displaystyle 18\displaystyle =\displaystyle 3xSubtract 2x from both sides
\displaystyle x\displaystyle =\displaystyle \dfrac{18}{3}Divide both sides by 3
\displaystyle x\displaystyle =\displaystyle 6Symmetric property of equality

Outcomes

MA.912.GR.6.1

Solve mathematical and real-world problems involving the length of a secant, tangent, segment or chord in a given circle.

MA.912.GR.6.2

Solve mathematical and real-world problems involving the measures of arcs and related angles.

MA.912.GR.6.3

Solve mathematical problems involving triangles and quadrilaterals inscribed in a circle.

MA.912.LT.4.8

Construct proofs, including proofs by contradiction.

MA.912.LT.4.10

Judge the validity of arguments and give counterexamples to disprove statements.

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