Circles

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Circles. Chapter 10. 10.1 Tangents to Circles. Circle : the set of all points in a plane that are equidistant from a given point. Center : the given point. Radius : a segment whose endpoints are the center of the circle and a point on the circle. Vocabulary.
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CirclesChapter 1010.1 Tangents to Circles
  • Circle: the set of all points in a plane that are equidistant from a given point.
  • Center: the given point.
  • Radius: a segment whose endpoints are the center of the circle and a point on the circle.
  • Vocabulary
  • Chord: a segment whose endpoints are points on the circle.
  • Diameter: a chord that passes through the center of the circle.
  • Secant: a line that intersects a circle in two points.
  • Tangent: a line in the plane of a circle that intersects the circle in exactly one point.
  • More Vocabulary
  • Congruent Circles: two circles that have the same radius.
  • Concentric Circles: two circles that share the same center.
  • Tangent Circles: two circles that intersect in one point.
  • Tangent Theorems
  • If a line is tangent to a circle, then it is perpendicular to the radius drawn to the point of tangency.
  • PQTangent Theorems
  • In a plane, if a line is perpendicular to a radius of a circle at its endpoint on the circle, then the line is tangent to the circle.
  • PQTangent Theorems
  • If two segments from the same exterior point are tangent to a circle then they are congruent.
  • QPExamples
  • Find an example for each term:
  • Center Chord Diameter Radius Point of Tangency Common external tangent Common internal tangent SecantExamples
  • The diameter is given. Find the radius.
  • d=15cm
  • d=6.7in
  • d=3ft
  • d=8cm
  • Examples
  • The radius is given. Find the diameter.
  • r = 26in
  • r = 62ft
  • r = 8.7in
  • r = 4.4cm
  • Examples
  • Tell whether AB is tangent to C.
  • A145B15CExamples
  • Tell whether AB is tangent to C.
  • A12C168BExamples
  • AB and AD are tangent to C. Find x.
  • D2x + 7A5x - 8B10.2 Arcs and Chords
  • An angle whose vertex is the center of a circle is a central angle.
  • If the measure of a central angle is less than 180 , then A, B and the points in the interior of APB form a
  • minor arc.
  • Likewise, if it is greater
  • than 180, if forms a major arc.Arcs and Chords
  • If the endpoints of an arc are the endpoints of a diameter, then the arc is a semicircle.
  • The measure of an arc is the same as the measure of its central angle.
  • Examples
  • Find the measure of each arc.
  • MN
  • MPN
  • PMN
  • Arc Addition Postulate
  • The measure of an arc formed by two adjacent arcs is the sum of the measures of the two arcs.
  • mABC = mAB + mBC
  • Examples
  • Find the measure of each arc.
  • GE
  • GEF
  • GF
  • Theorems
  • In the same circle, or in congruent circles, congruent chords have congruent arcs and congruent arcs have congruent chords.
  • In the same circle, or in congruent circles, two chords are congruent iff they are equidistant from the center.
  • Examples
  • Determine whether the arc is minor, major or a semicircle.
  • PQ
  • SU
  • QT
  • TUP
  • PUQ
  • Examples
  • KN and JL are diameters. Find the indicated measures.
  • mKL
  • mMN
  • mMKN
  • mJML
  • Examples
  • Find the value of x. Then find the measure of the red arc.
  • Homework
  • Pg. 600 # 26-28, 37, 39, 47, 48
  • Pg. 607 # 12-30 even, 32-34, 37-38
  • 10.3 Inscribed Angles
  • An inscribed angle is an angle whose vertex is on a circle and whose sides contain chords of the circle.
  • The arc that lies in the interior of an inscribed angle and has endpoints on the angle is called the intercepted arc.
  • Measure of an Inscribed Angle
  • If an angle is inscribed in a circle, then its measure is half the measure of its intercepted arc.
  • mADB = ½ mAB
  • Find the measure of the blue arc or angle
  • mQTS =
  • m NMP =
  • Theorem
  • If two inscribed angles if a circle intercept the same arc, then the angles are congruent.
  • Polygons and circles
  • If all of the vertices of a polygon lie on a circle, the polygon is inscribed in the circle, and the circle is circumscribed about the polygon.
  • Theorems
  • If a right triangle is inscribed in a circle, then the hypontenuse is a diameter of the circle.
  • B is a right angle iff AC is a diameter of the circle.
  • Theorems
  • A quadrilateral can be inscribed in a circle iff its opposite angles are supplementary.
  • D, E, F, and G lie on some circle, C, if and only if mD + mF = 180° and mE + mG = 180° Examples
  • Find the value of each variable.
  • More Examples
  • Pg 616 # 2-8
  • 10.4 Other Angle Relationships in Circles
  • If a tangent and a chord intersect at a point on a circle, then the measure of each angle formed is one half the measure of its intercepted arc.
  • m 1 = ½ mAB
  • m 2 = ½ mBCA
  • Examples
  • Line m is tangent to the circle. Find the measure of the red angle or arc.
  • Examples
  • BC is tangent to the circle. Find m CBD.
  • Theorems
  • If two chords intersect in the interior of a circle, then the measure of each angle is one half the sum of the measures of the arcs intercepted by the angle and its vertical angle.
  • x = ½ (mPS + mRQ)
  • x = ½ (106 + 174 )
  • x = 140
  • TheoremsBA
  • m 1 = ½ (mBC – mAC)
  • m 2 = ½ (mPQR – mPR)
  • m 3 = ½ (mXY – mWZ)
  • 1CP2QRXW3ZYMore Examples
  • Pg. 624 #2-7
  • 10.5 Segment Lengths in Circles
  • When 2 chords intersect inside of a circle, each chord is divided into 2 segments, called segments of a chord.
  • When this happens, the product of the lengths of the segments of one chord is equal to the product of the lengths of the segments of the other chord.
  • Examples
  • Find x.
  • Vocabulary
  • PS is a tangent segment because it is tangent to the circle at an endpoint.
  • PR is a secant segment and PQ is the external segment of PR.
  • Theorems
  • EA EB = EC ED
  • BAECDTheorems
  • (EA)2 = EC ED
  • EA is a tangent segment, ED is a secant segment.
  • AECDExamples
  • Find x.
  • Examples
  • Find x.
  • x ___ = 10 ___X2 = 4 ____Examples10.6 Equations of Circles
  • You can write the equation of a circle in a coordinate plane if you know its radius and the coordinates of its center.
  • Suppose the radius is r and its center is at ( h, k)
  • (x – h)2 + (y – k)2 = r2
  • (standard equation of a circle)
  • Examples
  • If the circle has a radius of 7.1 and a center at ( -4, 0), write the equation of the circle.
  • (x – h)2 + (y – k)2 = r2
  • (x – -4)2 + (y – 0)2 = 7.12
  • (x + 4)2 + y2 = 50.41
  • Examples
  • The point (1, 2) is on a circle whose center is (5, -1). Write the standard equation of the circle.
  • Find the radius. (Use the distance formula)
  • .
  • .
  • .
  • (x – 5)2 + (y – -1)2 = 52
  • (x – 5)2 + (y +1)2 = 25
  • Graphing a Circle
  • The equation of the circle is: (x + 2)2 + (y – 3)2 = 9
  • Rewrite the equation to find the center and the radius.
  • (x – (-2))2 + (y – 3)2 = 32
  • The center is (-2, 3) and the radius is 3.
  • Graphing a Circle
  • The center is (-2, 3) and the radius is 3.
  • Examples
  • Do Practice 10.6C or B together.
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