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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. ABC\begin{align*}\triangle ABC\end{align*} is an equilateral triangle in A\begin{align*}\bigodot A\end{align*}. Find mBCˆ\begin{align*}m\widehat{BC}\end{align*} and \begin{align*}m \widehat{BDC}\end{align*}.
4. \begin{align*}\triangle ABC\end{align*} and \begin{align*}\triangle ADE\end{align*} are equilateral triangles in \begin{align*}\bigodot A\end{align*}. 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, \begin{align*}A-E\end{align*}. \begin{align*}\bigodot A \cong \bigodot E\end{align*} and \begin{align*}\bigodot B \cong \bigodot C \cong \bigodot D\end{align*}. Each window is topped with a \begin{align*}240^\circ\end{align*} 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 \begin{align*}\overline{BC} \cong \overline{DE}\end{align*} and \begin{align*}\widehat{BC} \cong \widehat{DE}\end{align*}.

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, \begin{align*}\overline{BE} \cong \overline{CD}\end{align*} and \begin{align*}\widehat{BE} \cong \widehat{CD}\end{align*}.

In the second circle, \begin{align*}\triangle BAE \cong \triangle CAD\end{align*} by SAS.

Example 1: Use \begin{align*}\bigodot A\end{align*} to answer the following.

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

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

Solution:

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

b) \begin{align*}m \widehat{CD} \cong m \widehat{BD}\end{align*} because \begin{align*}BD = CD\end{align*}.

\begin{align*}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\end{align*}

Investigation 9-2: Perpendicular Bisector of a Chord

Tools Needed: paper, pencil, compass, ruler

1. Draw a circle. Label the center \begin{align*}A\end{align*}.

2. Draw a chord. Label it \begin{align*}\overline{BC}\end{align*}.

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

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

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

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

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

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 \begin{align*}x\end{align*} and \begin{align*}y\end{align*}.

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

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 \begin{align*}x\end{align*} and \begin{align*}y\end{align*}.

Solution: The diameter is perpendicular to the chord, which means it bisects the chord and the arc. Set up an equation for \begin{align*}x\end{align*} and \begin{align*}y\end{align*}.

\begin{align*}(3x-4)^\circ& =(5x-18)^\circ \qquad y+4=2y+1\\ 14^\circ& =2x \qquad \qquad \qquad \ \ \ 3=y\\ 7^\circ& =x \qquad \end{align*}

## 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 \begin{align*}\overline{AB}\end{align*} and \begin{align*}\overline{CD}\end{align*}. You can use Investigation 3-2
3. Erase the arc marks and lines beyond the points of intersection, leaving \begin{align*}\overline{FE}\end{align*} and \begin{align*}\overline{EG}\end{align*}. 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 \begin{align*}FE = EG\end{align*} and \begin{align*}\overline{EF} \perp \overline{EG}\end{align*}, then \begin{align*}\overline{AB}\end{align*} and \begin{align*}\overline{CD}\end{align*} are equidistant to the center and \begin{align*}\overline{AB} \cong \overline{CD}\end{align*}.

Example 5: Algebra Connection Find the value of \begin{align*}x\end{align*}.

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

\begin{align*}6x-7& = 35\\ 6x& = 42\\ x& =7\end{align*}

Example 6: \begin{align*}BD = 12\end{align*} and \begin{align*}AC = 3\end{align*} in \begin{align*}\bigodot A\end{align*}. Find the radius.

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

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

Example 7: Find \begin{align*}m\widehat{BD}\end{align*} from Example 6.

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

\begin{align*}\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}\end{align*}

Know What? Revisited In the picture, the chords from \begin{align*}\bigodot A\end{align*} and \begin{align*}\bigodot E\end{align*} are congruent and the chords from \begin{align*}\bigodot B, \ \bigodot C,\end{align*} and \begin{align*}\bigodot D\end{align*} 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. \begin{align*}\underline{\;\;\;\;\;\;\;\;\;} \cong \overline{DF}\end{align*}
2. \begin{align*}\widehat{AC} \cong \underline{\;\;\;\;\;\;\;\;\;}\end{align*}
3. \begin{align*}\widehat{DJ} \cong \underline{\;\;\;\;\;\;\;\;\;}\end{align*}
4. \begin{align*}\underline{\;\;\;\;\;\;\;\;\;} \cong \overline{EJ}\end{align*}
5. \begin{align*}\angle AGH \cong \underline{\;\;\;\;\;\;\;\;\;}\end{align*}
6. \begin{align*}\angle DGF \cong \underline{\;\;\;\;\;\;\;\;\;}\end{align*}
7. List all the congruent radii in \begin{align*}\bigodot G\end{align*}.

Find the value of the indicated arc in \begin{align*}\bigodot A\end{align*}.

1. \begin{align*}m \widehat{BC}\end{align*}
2. \begin{align*}m\widehat{BD}\end{align*}
3. \begin{align*}m\widehat{BC}\end{align*}
4. \begin{align*}m\widehat{BD}\end{align*}
5. \begin{align*}m\widehat{BD}\end{align*}
6. \begin{align*}m\widehat{BD}\end{align*}

Algebra Connection Find the value of \begin{align*}x\end{align*} and/or \begin{align*}y\end{align*}.

1. \begin{align*}AB = 32\end{align*}
2. \begin{align*}AB = 20 \end{align*}
3. Find \begin{align*}m\widehat{AB}\end{align*} in Question 20. Round your answer to the nearest tenth of a degree.
4. Find \begin{align*}m\widehat{AB}\end{align*} 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 \begin{align*}A\end{align*} is the center of the circle.

1 & 2. Answers will vary

3. \begin{align*}m\widehat{BC}=60^\circ, m\widehat{BDC}=300^\circ\end{align*}

4. \begin{align*}\overline{BC} \cong \overline{DE}\end{align*} and \begin{align*}\widehat{BC} \cong \widehat{DE}\end{align*}

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