<meta http-equiv="refresh" content="1; url=/nojavascript/">
You are reading an older version of this FlexBook® textbook: CK-12 Geometry - Basic Go to the latest version.

# 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.

1 & 2. Answers will vary

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

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

8 , 9 , 10

Feb 22, 2012

Dec 11, 2014