10. Circles

Lesson

A chord is a line segment with endpoints on the circumference of a circle. The diameter is the longest chord of a circle, and it divides the circle into two semi-circles. All other chords divide the circle into a major arc and minor arc. A minor arc has a corresponding central angle of less than $180^\circ$180°, while the major arc has corresponding central angle of greater than $180^\circ$180°.

If two chords of a circle are congruent, what can we conclude about the arcs of those chords?

1. Using the applet below, move points $C$`C` and $D$`D` to change the lengths of the chords. Move point $E$`E` to change the location of $\overline{EF}$`E``F` around the circle. Move point $B$`B` to change the size of the circle.

2. Is it always true that the arcs $EF$`E``F` and $CD$`C``D` are the same length? Explain your reasoning (hint: click the checkbox to show the circle radii).

The applet above demonstrates following theorem relating the chords and arcs of a circle.

Theorem

In the same circle or in congruent circles, two arcs are congruent if and only if their corresponding chords are congruent.

As the applet alluded to, we can prove this using congruent triangles. Since we know that congruent arcs are always formed by congruent central angles, we can deduce the following corollary to our theorem.

Corollary

In the same circle or in congruent circles, chords are congruent if and only if their corresponding central angles are congruent.

If a line, line segment, or ray divides an arc into two congruent arcs, then we say it bisects the arc.

A special phenomenon happens when the radius of a circle is perpendicular to a chord. Use the applet below to test create and test a conjecture. The radius $\overline{AG}$`A``G` is perpendicular to the chord $\overline{CD}$`C``D`, so what does it do to the chord $\overline{CD}$`C``D` and the minor arc of $CD$`C``D`?

If you hypothesized that the radius bisects the arc and the chord, you were correct. We can prove our conjecture using what we know of congruent triangles (hint: use the checkbox in the applet to show the radii of the circle). Since we can prove our conjecture, it's a theorem!

Theorem

If a diameter or radius of a circle is perpendicular to a chord, then it bisects the chord and its arc.

The converse of the theorem can also be proven using congruent triangles.

Converse

The perpendicular bisector of a chord is a diameter (or radius) of the circle.

We can also see another phenomenon relating two chords and their distance from the center of the circle. The following theorem can also be proven using congruent triangles.

Theorem

In the same circle or in congruent circles, two chords are congruent if and only if they are equidistant from the center.

$C$`C` is the center of the circle. Calculate $x$`x`.

Find the length of $\overline{AB}$`A``B` in circle $O$`O`.

What is the length of $x$`x`?