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9.3: Properties of Chords

Difficulty Level: At Grade Created by: CK-12

Learning Objectives

  • Find the lengths of chords in a circle.
  • Discover properties of chords and arcs.

Review Queue

  1. Draw a chord in a circle.
  2. Draw a diameter in the circle from #1. Is a diameter a chord?
  3. \triangle ABC is an equilateral triangle in \bigodot A. Find m\widehat{BC} and m \widehat{BDC}.
  4. \triangle ABC and \triangle ADE are equilateral triangles in \bigodot A. List a pair of congruent arcs and chords.

Know What? To the right is the Gran Teatro Falla, in Cadiz, Andalucía, Spain. Notice the five windows, A-E. \bigodot A \cong \bigodot E and \bigodot B \cong \bigodot C \cong \bigodot D. Each window is topped with a 240^\circ arc. The gold chord in each circle connects the rectangular portion of the window to the circle. Which chords are congruent?

Recall from the first section, a chord is a line segment whose endpoints are on a circle. A diameter is the longest chord in a circle.

Congruent Chords & Congruent Arcs

From #4 in the Review Queue above, we noticed that \overline{BC} \cong \overline{DE} and \widehat{BC} \cong \widehat{DE}.

Theorem 9-3: In the same circle or congruent circles, minor arcs are congruent if and only if their corresponding chords are congruent.

In both of these pictures, \overline{BE} \cong \overline{CD} and \widehat{BE} \cong \widehat{CD}.

In the second circle, \triangle BAE \cong \triangle CAD by SAS.

Example 1: Use \bigodot A to answer the following.

a) If m \widehat{BD} = 125^\circ, find m \widehat{CD}.

b) If m \widehat{BC} = 80^\circ, find m \widehat{CD}.

Solution:

a) BD = CD, which means the arcs are equal too. m \widehat{CD} = 125^\circ.

b) m \widehat{CD} \cong m \widehat{BD} because BD = CD.

m\widehat{BC} + m \widehat{CD} + m\widehat{BD} & =360^\circ\\80^\circ+2m\widehat{CD}& =360^\circ\\2m\widehat{CD} & = 280^\circ\\m\widehat{CD} & = 140^\circ

Investigation 9-2: Perpendicular Bisector of a Chord

Tools Needed: paper, pencil, compass, ruler

1. Draw a circle. Label the center A.

2. Draw a chord. Label it \overline{BC}.

3. Find the midpoint of \overline{BC} using a ruler. Label it D.

4. Connect A and D to form a diameter. How does \overline{AD} relate to \overline{BC}?

Theorem 9-4: The perpendicular bisector of a chord is also a diameter.

If \overline{AD} \perp \overline{BC} and \overline{BD} \cong \overline{DC} then \overline{EF} is a diameter.

If \overline{EF} \perp \overline{BC}, then \overline{BD} \cong \overline{DC} and \widehat{BE} \cong \widehat{EC}.

Theorem 9-5: If a diameter is perpendicular to a chord, then the diameter bisects the chord and its corresponding arc.

Example 2: Find the value of x and y.

Solution: The diameter perpendicular to the chord. From Theorem 9-5, x = 6 and y = 75^\circ.

Example 3: Is the converse of Theorem 9-4 true?

Solution: The converse of Theorem 9-4 would be: A diameter is also the perpendicular bisector of a chord. This is not true, a diameter cannot always be a perpendicular bisector to every chord. See the picture.

Example 4: Algebra Connection Find the value of x and y.

Solution: The diameter is perpendicular to the chord, which means it bisects the chord and the arc. Set up an equation for x and y.

(3x-4)^\circ& =(5x-18)^\circ \qquad y+4=2y+1\\14^\circ& =2x \qquad \qquad \qquad \ \ \ 3=y\\7^\circ& =x \qquad

Equidistant Congruent Chords

Investigation 9-3: Properties of Congruent Chords

Tools Needed: pencil, paper, compass, ruler

  1. Draw a circle with a radius of 2 inches and two chords that are both 3 inches. Label like the picture to the right. This diagram is drawn to scale.
  2. From the center, draw the perpendicular segment to \overline{AB} and \overline{CD}. You can use Investigation 3-2
  3. Erase the arc marks and lines beyond the points of intersection, leaving \overline{FE} and \overline{EG}. Find the measure of these segments. What do you notice?

Theorem 9-6: In the same circle or congruent circles, two chords are congruent if and only if they are equidistant from the center.

The shortest distance from any point to a line is the perpendicular line between them.

If FE = EG and \overline{EF} \perp \overline{EG}, then \overline{AB} and \overline{CD} are equidistant to the center and \overline{AB} \cong \overline{CD}.

Example 5: Algebra Connection Find the value of x.

Solution: Because the distance from the center to the chords is congruent and perpendicular to the chords, the chords are equal.

6x-7& = 35\\6x& = 42\\x& =7

Example 6: BD = 12 and AC = 3 in \bigodot A. Find the radius.

Solution: First find the radius. \overline{AB} is a radius, so we can use the right triangle \triangle ABC, so \overline{AB} is the hypotenuse. From Theorem 9-5, BC = 6.

3^2+6^2& =AB^2\\9+36&=AB^2\\AB&=\sqrt{45}=3\sqrt{5}

Example 7: Find m\widehat{BD} from Example 6.

Solution: First, find the corresponding central angle, \angle BAD. We can find m \angle BAC using the tangent ratio. Then, multiply m\angle BAC by 2 for m\angle BAD and m\widehat{BD}.

\tan^{-1} \left ( \frac{6}{3} \right ) & = m\angle BAC\\m\angle BAC & \approx 63.43^\circ\\m\angle BAD & \approx 2 \cdot 63.43^\circ \approx 126.86^\circ \approx m\widehat{BD}

Know What? Revisited In the picture, the chords from \bigodot A and \bigodot E are congruent and the chords from \bigodot B, \ \bigodot C, and \bigodot D are also congruent. We know this from Theorem 9-3.

Review Questions

  • Questions 1-3 use the theorems from this section and similar to Example 3.
  • Questions 4-10 use the definitions and theorems from this section.
  • Questions 11-16 are similar to Example 1 and 2.
  • Questions 17-25 are similar to Examples 2, 4, 5, and 6.
  • Questions 26 and 27 are similar to Example 7.
  • Questions 28-30 use the theorems from this section.
  1. Two chords in a circle are perpendicular and congruent. Does one of them have to be a diameter? Why or why not?
  2. Write the converse of Theorem 9-5. Is it true? If not, draw a counterexample.
  3. Write the original and converse statements from Theorem 9-3 and Theorem 9-6.

Fill in the blanks.

  1. \underline{\;\;\;\;\;\;\;\;\;} \cong \overline{DF}
  2. \widehat{AC} \cong \underline{\;\;\;\;\;\;\;\;\;}
  3. \widehat{DJ} \cong \underline{\;\;\;\;\;\;\;\;\;}
  4. \underline{\;\;\;\;\;\;\;\;\;} \cong \overline{EJ}
  5. \angle AGH \cong \underline{\;\;\;\;\;\;\;\;\;}
  6. \angle DGF \cong \underline{\;\;\;\;\;\;\;\;\;}
  7. List all the congruent radii in \bigodot G.

Find the value of the indicated arc in \bigodot A.

  1. m \widehat{BC}
  2. m\widehat{BD}
  3. m\widehat{BC}
  4. m\widehat{BD}
  5. m\widehat{BD}
  6. m\widehat{BD}

Algebra Connection Find the value of x and/or y.

  1. AB = 32
  2. AB = 20
  3. Find m\widehat{AB} in Question 20. Round your answer to the nearest tenth of a degree.
  4. Find m\widehat{AB} in Question 25. Round your answer to the nearest tenth of a degree.

In problems 28-30, what can you conclude about the picture? State a theorem that justifies your answer. You may assume that A is the center of the circle.

Review Queue Answers

1 & 2. Answers will vary

3. m\widehat{BC}=60^\circ, m\widehat{BDC}=300^\circ

4. \overline{BC} \cong \overline{DE} and \widehat{BC} \cong \widehat{DE}

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CK.MAT.ENG.SE.1.Geometry-Basic.9.3

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