1. *If a radius of a circle is perpendicular to a chord, then the radius bisects the chord.*

Here's a graphical representation of this theorem:

2. *In a circle or in congruent circles, if two chords are the same distance from the center, then they are congruent*.

Using these theorems in action is seen in the example below:

**1. Problem: Find ***CD*.
Given: *Circle R* is congruent to *circle S*.
*Chord AB* = 8.
*RM = SN*.
**Solution:** By theorem number 2 above, *segment AB* is
congruent to *segment CD*. Therefore, *CD*
equals **8**.

** Oh, the wonderfully confusing world of geometry! :-) The tangent being discussed here is ***not* the trigonometric ratio. This kind of tangent is a line or line segment that touches the perimeter of a circle at one point only and is perpendicular to the radius that contains the point.

**
1. Problem: Find the value of x.
Given: ***Segment AB* is tangent to
*circle C* at *B*.
Solution: *x* is a radius of the circle.
Since *x* contains *B*, and *AB*
is a tangent segment, *x* must be
perpendicular to *AB* (the definition of
a tangent tells us that).
If it is perpendicular, the triangle
formed by *x*, *AB*, and *CA* is a right
triangle.
Use the Pythagorean Theorem to
solve for x.
*15*^{2} + x^{2} = 17^{2}
x^{2} = 64
**x = 8**

** ***Congruent arcs* are arcs that have the same degree measure and are in the same circle or in congruent circles.

Arcs are very important and let us find out a lot about circles. Two theorems involving arcs and their central angles are outlined below.

1. * For a circle or for congruent circles, if two minor arcs are congruent, then their central angles are congruent.*

2. * For a circle or for congruent circles, if two central angles are congruent, then their arcs are congruent.*

Example:

** An ***inscribed angle* is an angle with its vertex on a circle and with sides that contain chords of the circle. The figure below shows an inscribed angle.

** The most important theorem dealing with inscribed angles is stated below. **

*The measure of an inscribed angle is equal to one-half the degree measure of its intercepted arc.*

Refine
By brindhamadhu
on the 20th of January, 2013