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Mth 97 Fall 2013 Chapter 7 Section 7.1 – Central Angles and Inscribed Angles An ___________ of a circle consists of the points A and B of the circle together with the portion of the circle contained between the two points. Points A and B are called the __________________ of the arc. A __________________________ is an arc whose endpoints are the endpoints of a diameter and is named by an arc symbol above three points: endpoint, another point on the arc between the endpoints, endpoint. An arc that is shorter than a semicircle is called a __________________ arc and is named by an arc symbol above just the endpoints. An arc that is longer than a semicircle is called a __________________ arc and is named by an arc symbol above three points: endpoint, another point on the arc between the endpoints, endpoint. E A D G B Minor arc AB Semicircle DEF Major arc GHI H F I Symbols A Central Angle is an angle whose vertex is the ______________ of the circle and whose sides intersect the circle. The measure of an arc of a central angle is equal to the measure of the angle. mAOB x if and only if AB = A O B Without using a protractor find the measures of the following arcs and central angles in circle O. 40° C AB = B 110° 55° O D ACD = BOC = DOE = CAD = AOE = AE = A E Postulate 7.1 – Length of an Arc – The ratio of the length of an arc l , of an arc of a circle to the circumference, C, of the circle equals the ratio of the measure of the central angle of the arc, x°, to 360°. r x° O l l x C 360 or l x 2 r 360 A grandfather clock has a pendulum 32 inches long. If the pendulum swings through and arc of 12°, how far does the pendulum travel? 1 Mth 97 Fall 2013 Chapter 7 Postulate 7.2 – Area of a Sector – The ratio of the area of a sector of a circle to the area of the circle is equal to the ratio of the measure of the central angle of the sector, x°, to 360°. B A A sector x x x or A sector Acircle r 2 360 360 A circle 360 r x° O Find the area of the minor sector if mAOB 75 and the radius is 10 cm. An ____________________ angle is an angle whose vertex lies on the circle and its sides each intersect the circle in another point. Inscribed Angle Theorem – The measure of an inscribed angle in a circle is equal to half the measure of its intercepted arc. A mABC B 1 AC 2 Find mABC if AC = 66°. C If AB = 140°, find mACB and mAOB . C O B mACB = mAOB = A Find y, if mTPR 55 If AB = 150° and BC = 110°, find x. A P x° B A y° T R C 2 Mth 97 Fall 2013 Chapter 7 Corollary 7.2 – Inscribed angles that intercept the same arc (or congruent arcs) are congruent. A mABD mACD B C 1 AD 2 If AD = 48°, find the measure of each angle. D mABD mACD Corollary 7.3 – An angle is inscribed in a semicircle if and only if it is a right angle. C A B O C is inscribed in semicircle ACB mC= Find the measure of the following angles where O is the center of the circle and AC and DF are diameters. The figure is not drawn to scale. 120° D C O A F mCOD mC mDBC mF mADF mDAF mDOA mDAB 60° B Do ICA 12, problems 1 and 2 3 Mth 97 Fall 2013 Chapter 7 Section 7.2 – Chords of a Circle A __________________ is a segment of a circle whose endpoints are on the circle. A B O In circle O, __________, __________, and _________ are chords. C D A _____________________ is a chord of the circle that goes through the center of the circle. ________ A ____________________ is half of a diameter of a diameter (from the center to a point on the circle). The plural of radius is ______________. In circle O _______, ________, and ________ radii. Theorems important for constructions: l A Theorem 7.4 – The perpendicular bisector of a chord contains the center of the circle. C Theorem 7.5 - The intersection of the perpendicular bisectors of any two nonparallel chords is the center of the circle. m B O D Using a compass and straight edge draw a circle with two nonparallel chords. Label the center of the circle P and your chords AB and CD . Construct the perpendicular bisector of each chord. (See the bottom of page 375 for different methods you can use to bisect a segment.) Corollary 7.6 – If two circles, O and P, intersect in two points A and B, then the line containing O and P is the perpendicular bisector of AB . AC = A P O C B 4 Mth 97 Fall 2013 Chapter 7 Measures of Angles formed by Chords Theorem 7.7 – If two chords intersect, then the measure of any one of the vertical angles formed is equal to half the sum of the measures of the two arcs intercepted by the two vertical angles. B A 1 AB CD mCED 2 mAED 1 AD BC mBEC 2 C E mAEB D If AD 50 and BC 30 , then mAED mBEC mAEB mCED Measures of Segments of Chords Theorem 7.8 – If two chords of a circle intersect the product of the lengths of the two segments formed on one chord is equal to the product of the lengths of the two segments formed on the other chord. A (AE)(EC) = (BE)(ED) E B If AE = 8, EC = 3, and BE = 4, find ED. C D If AE = 9.8 cm, CE = 5.7 cm, and DE = 3.8 cm, find BE. If AC = 10 cm, BE = 8 cm, DE = 3 cm, find AE. Use the circle below to find the following measures, if possible. A 40° B 6 8 E 16 mAEB mCEB mCED mAED EB AD C 60° D 5 Mth 97 Fall 2013 Chapter 7 More practice. Figures are not drawn to scale. If AB 24 and mAEB 35 , find CD . D E C A C If mCED = 41 and mDAC = 25 , find AB . D E B A B Section 7.3 – Secants and Tangents A _________________ line intersects a circle in two points. A ___________________ line intersects a circle in exactly one point, called the point of tangency. A B C Chord Secant m Tangent Point of tangency Angles formed by Secants and Tangents The measure of the angles formed outside of circles by secant lines, a secant and a tangent line, or two tangents is half the difference of the intercepted arcs. Theorem 7.9 – If two secant lines intersect outside a circle, the measure of the acute angle formed is half the difference of the measures of the intercepted arcs. B A E 1 D mE = O O 2 C Theorem 7.13 – If a secant and a tangent line intersect outside a circle, the measure of the angle formed is half the difference of the measures of the intercepted arcs. A B mB = 1 2 C D 6 Mth 97 Fall 2013 Chapter 7 Theorem 7.14 – If two tangent lines intersect outside a circle, the measure of the angle formed is half the difference of the measures of the intercepted arcs. A B mB = 1 2 C Theorem 7.11 – The radius or diameter of a circle is perpendicular to a tangent line at its point of tangency. B O A OA which means ____________ is a right angle. Theorem 7.12 – The measure of the angle formed when a chord intersects a tangent line at the point of tangency is half the measure of the arc intercepted by the chord and tangent line. A C mABC 1 2 B Measure of Segments formed by Secant and Tangent Lines A B E Theorem 7.10 – If two secants intersect outside a circle, then the product of the lengths of the two segments formed on one secant (vertex to point on the circle) is equal to the product of the lengths of the corresponding segments on the other secant. D C (AE)(BE) = A B C AB = A B Theorem 7.16 – If we draw a tangent and a secant line from the same point in the exterior of the circle, the length of the tangent segment is the mean proportional between the length of the external secant segment and the length of the secant segment inside the circle. D C BD Theorem 7.15 – If two tangent lines are drawn to a circle from the same point in the exterior of the circle, the distances from the common point to the points of tangency are equal. BC or (BA)(BA) = ( ) ( ) 7 Mth 97 Fall 2013 Chapter 7 Secant and Tangent Practice Find BC, mBD , and mCAE . Find the value of x, mBC , and mM . 120° A 6 C A 154 B 8 o 10 78o 90° D 18 E C x B 12 M 110° Find the value of x, mA , and AD. A Find the values of AC, x and y in circle below. D 4 E 5 A x° B 52° P y° 7 C x° D 84° 50° C 12 B P is the center of the circle. 8 Mth 97 Fall 2013 Chapter 7 Section 7.4 - Constructions Involving Circles Given an arc, construct the circle it is part of. First draw two nonparallel chords. Construct the perpendicular bisector of each chord to find the circle’s center. Set your compass for the radius and draw the rest of the circle. The circumscribed circle of a triangle is the unique circle that contains the triangle’s _______________. Constructing the circumscribed circle of a triangle First construct the perpendicular bisector of two sides of the triangle. Use the distance from the intersection point of the bisectors to any vertex as the radius to circumscribe the triangle in a circle. Theorem 7.17 – Circumcenter of a Triangle The perpendicular bisectors of the sides of a triangle ________________in a single point, the circumcenter. A P is the circumcenter of ∆ABC. P B C 9 Mth 97 Fall 2013 Chapter 7 Construct an altitude from each vertex to the opposite side. You may need to extend a side to help you construct the perpendicular. B Theorem 7.18 – Orthocenter of a Triangle The ______________ of a triangle intersect in a single point, the orthocenter. P is the orthocenter of ∆ABC. P A C The points A, B, C, and P form an orthocentric set which has the property that the triangle formed by any three of the four points in the set has the fourth point as its orthocenter. Constructing the inscribed circle of a triangle First construct any two angle bisectors of the triangle to locate the incenter, the point that is equidistant from all sides of the triangle. Next construct a perpendicular from the incenter to a side. Use the distance from the the incenter to the side as a radius to inscribe a circle within the triangle. 10 Mth 97 Fall 2013 Chapter 7 A Theorem 7.19 – Incenter of a Triangle The angle bisectors of a triangle meet in a single point, the incenter. P is the incenter of ∆ABC. P C B Constructing the centroid of a triangle First construct the midpoints of any two sides and draw the medians. Their intersection point is the centroid, or center of gravity, or balance point of the triangle. Theorem 7.20 – Centroid of a Triangle The medians of a triangle intersect in a single point, the centroid, which is two-thirds of the way from the vertex to the other endpoint of the median from that vertex. A P is the centroid of ∆ABC P C BP B 2 2 BF , AP AE 3 3 and CP 2 CD 3 Constructing a tangent to a circle First draw a ray from the center of the circle through the point on the circle you want the tangent to intersect. Next construct a line perpendicular to the radius you drew at the point 11