5.1: Chords, Arcs, and Angles | Geometry Honors

How it works

An inscribed angle is half the measure of its intercepted arc. A central angle equals its intercepted arc measure. A radius perpendicular to a chord bisects it; tangent segments from a common exterior point have equal lengths.

m\angle\text{inscribed}=\tfrac12 m\widehat{\text{intercepted arc}}

Worked examples

Example 1

An inscribed angle intercepts an arc of $\displaystyle 110^\circ$ . Find the angle.

The angle is $\displaystyle 110^\circ/2=55^\circ$ .

Example 2

A circle has radius $\displaystyle 5$ and a chord $\displaystyle 3$ units from the center. Find the chord length.

A perpendicular radius bisects the chord. Half its length is $\displaystyle \sqrt{25-9}=4$ , so the full chord is $\displaystyle 8$ .

Problems

Problem 5.1.1

A circle of radius $\displaystyle 5$ has a horizontal chord $\displaystyle 3$ units above its center. Find the chord length.

Circle radius 5 with a chord 3 units above the center; a perpendicular segment bisects it. — The perpendicular from the center bisects the chord.

Show worked solution

The right triangle from center to chord endpoint has hypotenuse $\displaystyle 5$ and vertical leg $\displaystyle 3$ . Half the chord is $\displaystyle 4$ , hence its length is $\displaystyle 8$ .

Circle radius 5 with a chord 3 units above the center; a perpendicular segment bisects it. The solution adds labeled relationships. — Half the chord is the leg of a 3-4-5 triangle, so the chord is 8.

Problem 5.1.2

A tangent from $\displaystyle P$ touches a radius- $\displaystyle 6$ circle at $\displaystyle T$ . If the center is $\displaystyle 10$ units from $\displaystyle P$ , find $\displaystyle PT$ .

Show worked solution

The radius is perpendicular to the tangent. Thus $\displaystyle PT=\sqrt{10^2-6^2}=8$ .

Problem 5.1.3

An inscribed angle intercepts a semicircle. Find its measure.

Show worked solution

A semicircle measures $\displaystyle 180^\circ$ , so the inscribed angle measures $\displaystyle 90^\circ$ .

Lesson Problems PDF Solutions PDF Unit MCQ

How it works

Worked examples

Example 1

Example 2

Problems

Problem 5.1.1

Problem 5.1.2

Problem 5.1.3

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