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Circles. 10-1 Circles. C. I. Definitions Circle The set of all points in a plane that are at a given distance from a given point in that plane . Symbol ○R Radius The distance between the center of a circle and any point on the edge of the circle. AR,BR

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## Circles

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**10-1 Circles**C • I. Definitions • Circle The set of all points in a plane that are at a given distance from a given point in that plane. • Symbol ○R • Radius The distance between the center of a circle and any point on the edge of the circle.AR,BR • Diameter The distance across the circle that goes through the radiusAB • Chord A segment that goes from one side of a circle to another. AC • Circumference the distance around the circleC= 2 r • Total degrees 360 B A R**II. EXAMPLES**• 1. Find the circumference if • a. r = 8 b. d = 12**3. Circle A radius=8, Circle B radius = 14, and JE =4.**Find EB and DC. D C J A E B**I. Central Angle**an angle whose vertex contains the center of a circle. 10-2 Angles and Arcs**II. Arc**Part of a circle; the curve between two points on a circle.**If circle is divided into two unequal parts or arcs, the**shorter arc (in red) is called the minor arc and the longer arc (in blue) is called the major arc. A Minor arc- 2 letters Major arc- 3 letters B C**III. Semicircle**• a semicircle is an arc that makes up half of a circle 180°**Arc Addition Postulate - The measures of an arc formed by**two adjacent arcs is the sum of the measures of the two arcs. That is, if B is a point on , then + = .**IV. ARC MEASURE**The measure of a minor arc = central angle. The measure of a major arc = 360 minus the measure of its central angle.**V. ARC LENGTH**LENGTH OF THE ARC is a part of the circumference proportional to the measure of the central angle when compared to the entire circle**VII. CONCENTRIC CIRCLES**CONCENTRIC CIRCLES lie in the same plane and have the same center, but have different radii. ALL CONCENTRIC CIRCLES ARE SIMILAR BC ALL CIRCLES ARE SIMILAR!**VIII. CONGRUENT ARCS**TWO ARCS WITH THE SAME MEASURE AND LENGTH**Example 1: Find the length of arc RT and the measurement in**degrees.**a. Find the length of arcs RT and RSTb. Find the**measurement in degrees of both.**3. Find the arc length of RT and the degrees measurement of**RT.**If <NGE < EGT, <AGJ =2x, <JGT = x + 12, and AT and JN**are diameters, find the following:a. x b. m NE c. m JNE A J E N T**6. Find x.**N M 9x 8x Q A 19x O R**I. Arc of the chord**When a minor arc and a chord share the same endpoints, we called the arc the ARC OF THE CHORD. 10-3 Arcs and Chords**II. Relationships**• 2 minor arcs are if their chords are .**If a diameter is perpendicular to a chord, it bisects the**chord and the arc.**1. Circle N has a radius of 36.5 cm. Radius is**perpendicular to chord FG, which is 53 cm long. • a. If m FG= 85, find m HG. • b. Find NZ.**2. Chords FG and LY are equidistant from the center. If**the radius of M is 32, find FG and BY. FG = 46.4 LY = 23.2**XZ= 12, UV = 8, WY is a diameter.Find the length of a**radius.**6. IF AB and DC are both parallel and congruent and MP = 7,**find PQ.**10-4 Inscribed Angles**• I. Definitions • Inscribed angle — An angle that has its vertex on the circle and its sides are chords of the circle • Intercepted arc — An intercepted arc is the arc that lies "inside" of an inscribed angle**If an angle is inscribed in a circle, then the measure of**the angle equals one-half the measure of its intercepted arc**If two inscribed angles of a circle or congruent circles**intercept congruent arcs or the same arc, then the angles are congruent**If an inscribed angle of a circle intercepts a semicircle,**then the angle is a right angle**If a quadrilateral is inscribed in a circle, then its**opposite angles are supplementary**3. If mLM=120, mMN=45, and mNQ=105, find the numbered**angles. < 1= 22.5 < 2 = 60 < 3 = 45 < 4 = 22.5 < 5 = 112.5**4. If <2= 3a + 2 and < 3= 12 a – 2, find the measures of**the numbered angles m1 = 45, m4 = 45 <2 = 20, < 3= 70**5. If mW = 74 and mZ = 112, find mY and mX.**• = mX • 106 = mY**10-5 Tangents**• I. A line is TANGENT to a circle if it intersects the circle in EXACTLY ONE point. This point is called the POINT OF TANGENCY.**If a line is tangent to a circle, then it is perpendicular**to the radius drawn to the point of tangency.**II. Common External Tangents & Common Internal Tangents**A line or line segment that is tangent to two circles in the same plane is called a common tangent COMMON EXTERIOR**COMMON**INTERIOR**If two segments from the same exterior point are tangentto**the circle, then they are congruent

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