There are several important theorems about chords that will help you to analyze circles better.
1. Chord Theorem #1: 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, and .
2. Chord Theorem #2: The perpendicular bisector of a chord is also a diameter.
If and then is a diameter.
3. Chord Theorem #3: If a diameter is perpendicular to a chord, then the diameter bisects the chord and its corresponding arc.
If , then and .
4. Chord Theorem #4: 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 and , then and are equidistant to the center and .
What if you were given a circle with two chords drawn through it? How could you determine if these two chords were congruent?
Find the value of and .
The diameter is perpendicular to the chord, which means it bisects the chord and the arc. Set up equations for and .
and in . Find the radius.
First find the radius. is a radius, so we can use the right triangle with hypotenuse . From Chord Theorem #3, .
Use to answer the following.
- If , find .
, which means the arcs are congruent too. .
- If , find .
Find the values of and .
The diameter is perpendicular to the chord. From Chord Theorem #3, and .
Find the value of .
Because the distance from the center to the chords is equal, the chords are congruent.
Fill in the blanks.
- List all the congruent radii in .
Find the value of the indicated arc in .
Find the value of and/or .
- Find in Question 17. Round your answer to the nearest tenth of a degree.
- Find in Question 22. Round your answer to the nearest tenth of a degree.
In problems 25-27, what can you conclude about the picture? State a theorem that justifies your answer. You may assume that is the center of the circle.
To see the Review answers, open this PDF file and look for section 9.4.