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. 152 + x2 = 172 x2 = 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.
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.
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