What if you were given a circle with two chords drawn through it? How could you determine if these two chords were congruent? After completing this Concept, you'll be able to use four chord theorems to solve problems like this one.
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 .
Use to answer the following.
a) If , find .
b) If , find .
a) , which means the arcs are congruent too. .
b) because .
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.
1. Find the value of and .
2. and in . Find the radius.
3. Find from #2.
1. The diameter is perpendicular to the chord, which means it bisects the chord and the arc. Set up equations for and .
2. First find the radius. is a radius, so we can use the right triangle with hypotenuse . From Chord Theorem #3, .
3. First, find the corresponding central angle, . We can find using the tangent ratio. Then, multiply by 2 for and .
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.