Parallel Lines        Congruent Tri.        Congruent R. Tri.        Isosc. and Equil.        Quadrilaterals        Parallelograms        Ratios        Similar Polygons        Special Triangles   Circles        Area        Coordinate Geo.        Triangle Ineq.        Solids        Computer Fun Circles, the perfect shape!  On this page we hope to clear up problems that you might have with circles and the figures, such as radii, associated with them.  Just start scrolling down or click one of the links below to start understanding circles better! Chords & radii Tangents Congruent arcs Inscribed angles Angles involving tangents and/or secants Segments in circles Circumference and arc length Quiz on Circles All the "parts" of a circle, such as the radius, the diameter, etc., have a relationship with the circle or another "part" that can always be expressed as a theorem.  The two theorems that deal with chords and radii (plural of radius) are outlined below. 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. 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. ```1. Problem: Find the measure of each arc or angle listed below. arc QSR angle Q angle R Solution: Arc QSR is 180o because it is twice the measure of its inscribed angle (angle QPR, which is 90o). Angle Q is 60o because it is half of its intercepted arc, which is 120o. Angle R is 30o by the Triangle Sum Theorem which says a triangle has three angles which have measures that equal 180o when added together.``` In the last problem's figure, you noticed that angle P is inscribed in semicircle QPR and angle P = 90o.  This leads us to our next theorem, which is stated below. Any angle inscribed in a semicircle is a right angle. The one last theorem dealing with inscribed angles is a bit more complicated because it deals with quadrilaterals, too.  It is stated below. If a quadrilateral is inscribed in a circle, then both pairs of opposite angles are supplementary. ``` 1. Problem: Find the measure of arc GDE. Solution: By the theorem stated above, angle D and angle F are supplementary. Therefore, angle F equals 95o. The first theorem discussed in this section tells us the measure of an arc is twice that of its inscribed angle. With that theorem, arc GDE is 190o.``` When two secants intersect inside a circle, the measure of each angle formed is related to one-half the sum of the measures of the intercepted arcs.  The figure below shows this theorem in action. In the figure, arc AB and arc CD are 60o and 50o, respectively.  By the above stated theorem, the measures of both angle 1 and angle 2 in the figure are 55o. Sometimes, secants intersect outside of circles.  When this happens, the measure of the angle formed is equal to one-half the difference of the degree measures of the intercepted arcs. ```1. Problem: Find the measure of angle 1. Givens: Arc AB = 60o Arc CD = 100o Solution: By the theorem stated above, the measure of angle 1 = .5((arc CD) - (arc AB)) angle 1 = .5((100 - 60)) angle 1 = 20o``` Another way secants can intersect in circles is if they are only in line segments.  There is a theorem that tells us when two chords intersect inside a circle, the product of the measures of the two segments of one chord is equal to the product of the measures of the two segments of the other chord.  In the figure below, chords PR and QS intersect.  By the theorem stated above, PT * TR = ST * TQ. One last thing that has to be discussed when dealing with circles is circumference, or the distance around a circle.  The circumference of a circle equals 2 times PI times the measure of the radius.  That postulate is usually represented by the following equation (where C represents circumference and r stands for radius): C = 2(PI)r. For example, if a circle has a radius of 3, the circumference of the circle is 6(PI). Also, you can find the length of any arc when you know its degree measure and the measure of a radius with the following formula (L = length, n = degree measure of arc, r = radius): L = (n/360)(2(PI)r). ``` 1. Problem: Find the length of a 24o arc of a circle with a 5 cm radius. Solution: n 24 2(PI) L = ---(2(PI)r) = ---(2(PI))5 = ----- 360 360 3 The length of the arc is (2/3)(PI) cm. ``` Take the Quiz on circles.  (Very useful to review or to see if you've really got this topic down.)  Do it!

Math for Morons Like Us - Geometry: Circles
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