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9.3: Properties of Chords

Difficulty Level: At Grade Created by: CK-12
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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. ABC is an equilateral triangle in A. Find mBCˆ and mBDCˆ.
  4. ABC and ADE are equilateral triangles in 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, AE. AE and BCD. Each window is topped with a 240 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 BC¯¯¯¯¯¯¯¯DE¯¯¯¯¯¯¯¯ and BCˆDEˆ.

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, BE¯¯¯¯¯¯¯¯CD¯¯¯¯¯¯¯¯ and BEˆCDˆ.

In the second circle, BAECAD by SAS.

Example 1: Use A to answer the following.

a) If mBDˆ=125, find mCDˆ.

b) If mBCˆ=80, find mCDˆ.

Solution:

a) BD=CD, which means the arcs are equal too. mCDˆ=125.

b) mCDˆmBDˆ because BD=CD.

mBCˆ+mCDˆ+mBDˆ80+2mCDˆ2mCDˆmCDˆ=360=360=280=140

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 BC¯¯¯¯¯¯¯¯.

3. Find the midpoint of BC¯¯¯¯¯¯¯¯ using a ruler. Label it D.

4. Connect A and D to form a diameter. How does AD¯¯¯¯¯¯¯¯ relate to BC¯¯¯¯¯¯¯¯?

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

If AD¯¯¯¯¯¯¯¯BC¯¯¯¯¯¯¯¯ and BD¯¯¯¯¯¯¯¯DC¯¯¯¯¯¯¯¯ then EF¯¯¯¯¯¯¯¯ is a diameter.

If EF¯¯¯¯¯¯¯¯BC¯¯¯¯¯¯¯¯, then BD¯¯¯¯¯¯¯¯DC¯¯¯¯¯¯¯¯ and BEˆECˆ.

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.

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.

(3x4)147=(5x18)y+4=2y+1=2x   3=y=x

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 AB¯¯¯¯¯¯¯¯ and CD¯¯¯¯¯¯¯¯. You can use Investigation 3-2
  3. Erase the arc marks and lines beyond the points of intersection, leaving FE¯¯¯¯¯¯¯¯ and 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 EF¯¯¯¯¯¯¯¯EG¯¯¯¯¯¯¯¯, then AB¯¯¯¯¯¯¯¯ and CD¯¯¯¯¯¯¯¯ are equidistant to the center and AB¯¯¯¯¯¯¯¯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.

6x76xx=35=42=7

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

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

32+629+36AB=AB2=AB2=45=35

Example 7: Find mBDˆ from Example 6.

Solution: First, find the corresponding central angle, BAD. We can find mBAC using the tangent ratio. Then, multiply mBAC by 2 for mBAD and mBDˆ.

tan1(63)mBACmBAD=mBAC63.43263.43126.86mBDˆ

Know What? Revisited In the picture, the chords from A and E are congruent and the chords from B, C, and 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. DF¯¯¯¯¯¯¯¯
  2. ACˆ
  3. DJˆ
  4. EJ¯¯¯¯¯¯¯
  5. AGH
  6. DGF
  7. List all the congruent radii in G.

Find the value of the indicated arc in A.

  1. mBCˆ
  2. mBDˆ
  3. mBCˆ
  4. mBDˆ
  5. mBDˆ
  6. mBDˆ

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

  1. AB=32
  2. AB=20
  3. Find mABˆ in Question 20. Round your answer to the nearest tenth of a degree.
  4. Find mABˆ 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. mBCˆ=60,mBDCˆ=300

4. BC¯¯¯¯¯¯¯¯DE¯¯¯¯¯¯¯¯ and BCˆDEˆ

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