Arcs Chords And Central Angles

6.12: Chords and Fundamental Angle Arcs

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Arcs adamant by angles whose vertex is the centre of a circle and chords (segments that connect two points on a circumvolve).

Chords in Circles

Chord Theorems

At that place are several important theorems virtually chords that volition aid you lot to analyze circles better.

1. Chord Theorem #1: In the same circle or congruent circles, minor arcs are congruent if and merely if their respective chords are congruent.

Effigy \(\PageIndex{1}\)

In both of these pictures, \(\overline{Exist}\cong \overline{CD}\) and \(\widehat{Be}\cong \widehat{CD}\).

2. Chord Theorem #2: The perpendicular bisector of a chord is also a diameter.

Figure \(\PageIndex{2}\)

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

3. Chord Theorem #three: If a diameter is perpendicular to a chord, then the diameter bisects the chord and its corresponding arc.

Figure \(\PageIndex{three}\)

If \(\overline{EF}\perp \overline{BC}\), then \(\overline{BD}\cong \overline{DC}\)

4. Chord Theorem #4: In the aforementioned circumvolve or congruent circles, 2 chords are congruent if and simply if they are equidistant from the heart.

Figure \(\PageIndex{4}\)

The shortest distance from any point to a line is the perpendicular line between them. If \(Atomic number 26=EG\) and \(\overline{EF}\perp \overline{EG}\), and then \(\overline{AB}\) and \(\overline{CD}\) are equidistant to the center and\(\overline{AB}\cong \overline{CD}\).

What if yous were given a circumvolve with 2 chords drawn through information technology? How could you make up one's mind if these ii chords were congruent?

Example \(\PageIndex{ane}\)

Find the value of \(x\) and \(y\).

Figure \(\PageIndex{v}\)

Solution

The bore is perpendicular to the chord, which ways it bisects the chord and the arc. Set upwards equations for \(ten\) and \(y\).

\(\begin{assortment}{rlr}
(three x-four)^{\circ} & =(5 ten-18)^{\circ} & y+four=2 y+i \\
xiv & =2 x & 3=y \\
7 & =x
\stop{array}\)

Instance \(\PageIndex{2}\)

\(BD=12\) and \(AC=three\) in \(\bigodot A\). Notice the radius.

Effigy \(\PageIndex{6}\)

Solution

Start find the radius. \(\overline{AB}\) is a radius, and then nosotros can use the right triangle \Delta ABC\) with hypotenuse \(\overline{AB}\). From Chord Theorem #3, \(BC=six\).

\(\begin{aligned} iii^2+6^2&=AB^two \\ ix+36&=AB^2 \\ AB&=\sqrt{45}=3\sqrt{v}\end{aligned}\)

Instance \(\PageIndex{iii}\)

Use \(\bigodot A\) to respond the following.

Figure \(\PageIndex{7}\)

1. If \(k\widehat{BD}=125^{\circ}\), find \(k\widehat{CD}\).
2. If \(m\widehat{BC}=80^{\circ}\), notice \(chiliad\widehat{CD}\).

Solution

1. \(BD=CD\), which means the arcs are congruent too. \(m\widehat{CD}=125^{\circ}\).
2. \(k\widehat{CD}\cong m\widehat{BD}\) considering \(BD=CD\).

\(\begin{aligned} m\widehat{BC}+thousand\widehat{CD}+k\widehat{BD}&=360^{\circ} \\ 80^{\circ}+2m\widehat{CD}&=360^{\circ} \\ 2m\widehat{CD}&=280^{\circ} \\ thousand\widehat{CD}=140^{\circ}\end{aligned}\)

Example \(\PageIndex{4}\)

Detect the values of \(x\) and \(y\).

Figure \(\PageIndex{eight}\)

Solution

The diameter is perpendicular to the chord. From Chord Theorem #iii, \(x=6\) and \(y=75^{\circ}\).

Example \(\PageIndex{5}\)

Observe the value of \(x\).

Effigy \(\PageIndex{9}\)

Solution

Considering the distance from the centre to the chords is equal, the chords are congruent.

\(\begin{aligned} 6x−7&=35 \\ 6x&=42 \\ x&=seven \cease{aligned}\)

Review (Answers)

To meet the Review answers, open this PDF file and look for section 9.4.

Vocabulary

Term	Definition
chord	A line segment whose endpoints are on a circumvolve.
circle	The set of all points that are the same distance away from a specific point, called the center .
diameter	A chord that passes through the center of the circle. The length of a diameter is two times the length of a radius.
radius	The distance from the eye to the outer rim of a circle.

Additional Resources

Video: Chords in Circles Principles - Basic

Activities: Chords in Circles Discussion Questions

Study Aids: Circles: Segments and Lengths Study Guide

Do: Chords and Central Angle Arcs

Arcs Chords And Central Angles,

Source: https://k12.libretexts.org/Bookshelves/Mathematics/Geometry/06%3A_Circles/6.12%3A_Chords_and_Central_Angle_Arcs

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