- Find the lengths of chords in a circle.
- Discover properties of chords and arcs.
- Draw a chord in a circle.
- Draw a diameter in the circle from #1. Is a diameter a chord?
△ABC is an equilateral triangle in ⨀A. Find mBCˆ and mBDCˆ.
△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, A−E. ⨀A≅⨀E and ⨀B≅⨀C≅⨀D. 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 BCˆ≅DEˆ.
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 BEˆ≅CDˆ.
In the second circle, △BAE≅△CAD by SAS.
Example 1: Use ⨀A to answer the following.
a) If mBDˆ=125∘, find mCDˆ.
b) If mBCˆ=80∘, find mCDˆ.
a) BD=CD, which means the arcs are equal too. mCDˆ=125∘.
b) mCDˆ≅mBDˆ because BD=CD.
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 BEˆ≅ECˆ.
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.
Equidistant Congruent Chords
Investigation 9-3: Properties of Congruent Chords
Tools Needed: pencil, paper, compass, ruler
- 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.
- From the center, draw the perpendicular segment to AB¯¯¯¯¯¯¯¯ and CD¯¯¯¯¯¯¯¯. You can use Investigation 3-2
- 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.
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.
Example 7: Find mBDˆ from Example 6.
Solution: First, find the corresponding central angle, ∠BAD. We can find m∠BAC using the tangent ratio. Then, multiply m∠BAC by 2 for m∠BAD and mBDˆ.
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.
- 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.
- Two chords in a circle are perpendicular and congruent. Does one of them have to be a diameter? Why or why not?
- Write the converse of Theorem 9-5. Is it true? If not, draw a counterexample.
- Write the original and converse statements from Theorem 9-3 and Theorem 9-6.
Fill in the blanks.
- List all the congruent radii in ⨀G.
Find the value of the indicated arc in ⨀A.
Algebra Connection Find the value of x and/or y.
- Find mABˆ in Question 20. Round your answer to the nearest tenth of a degree.
- 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
4. BC¯¯¯¯¯¯¯¯≅DE¯¯¯¯¯¯¯¯ and BCˆ≅DEˆ