Lesson 7.1: Central Angles

Transcription

1 Lesson 7.1: Central Angles Definition 5.1 Arc An arc is a part of a circle. Types of Arc 1. Minor Arc 2. Major Arc 3. Semicircle Figure 5.1 Definition 5.2 Central Angle A central angle of a circle is an angle whose vertex is the center of the circle. Definition 5.3 Minor Arc Minor arc consists of points M and I and all points of O that are in the interior of central angle MOI. Figure 5.2

2 Definition 5.4 Major Arc Major arc consists of points M and I and all points of O that are in the exterior of central angle MOI. Figure 5.3 Definition 5.5 Semicircle Semicircle consists of endpoints D and M of diameter DM and all points O that lie on one side of DM. Figure 5.4 Note: The degree measure of an arc is defined in terms of its central angle. Definition 5.6 Degree Measure of Minor The degree measure of minor MI is equal to the degree measure of central angle MOI. Definition 5.7 Degree Measure of Major The degree measure of major MNI is equal to 360 minus the degree measure of central angle MOI. Definition 5.8 Degree Measure of Semicircle The degree measure of semicircle DIM is equal to 180.

3 Definition 5.9 Congruent Circles Congruent Circles are circles with the same radius. Definition 5.10 Congruent Arc Congruent Arcs are arcs with the same measures. Postulate 5.1 The Central Angle-Intercepted Arc Postulate (CA-IA Postulate) The measure of a central angle of a circle is equal to the measure of its intercepted arc. Refer to the figure and the given information to answer each. m CEI = 45 and bisects CER. Find the indicated measure: 1. mci 2. m IER and mir 3. m CER and mcr 4. mcli 5. m REL and mrl 6. mlcr Postulate 5.2 The Arc Addition Postulate Given point B on AC, then mac = mab + mbc. Postulate 5.3 A diameter divides a circle into two semicircles. Definition 5.11 Arc Length The measure of the central angle can also be used to determine the arc length. The arc length (or length of an arc) is different from the degree measure of an arc. That is, if a circle is made up of string, the length of the arc is the linear distance of the piece of string representing the arc. The length of the arc is a part of the circumference and proportional to the measure of the central angle when compared to the entire circle. In O, OY = 18 and m MOY = 60. Find the length of MY. Let's Practice: Direction: Use the figure at the right to identify each of the following major arcs 2. 2 minor arcs 3. an acute central angle 4. an obtuse central angle 5. a radius which is not a part of a diameter. 6. a semicircle

4 Direction: In E, LET TEI, m OEV = 5x, M VEI = x + 36, and OI and VL are diameters. Find each of the following. 1. x = 2. m OEV = 3. m VEI = 4. m TEI = 5. mlti = Direction: The circles at the right are concentric with center at L. Determine whether each statement is true or false. Justify your answer. 1. If mre = 40 and mno = 40, then RE NO. 2. If m ELR = 60, then mre = mno = LO LN 4. If mno = 50, then mear = If mre = mno, then RE NO.

5 Lesson 5.2: Inscribed Angle Definition 5.12 Inscribed Angle An inscribed angle in a circle is an angle whose vertex is on the circle and whose sides contain chords of the circle. Considering the definition, which among these three is/are inscribed angle? Figures 5.4 Theorem 5.1 The Inscribed Angle Theorem The measure of an inscribed angle is one-half the measure of its intercepted arc. Given: ARO is inscribed in D and AR contains D Prove: m ARO = 1 2 mao Using Theorem 5.1, in the figure, mbs = 64, find m BAS and m SOB. Theorem 5.2 The Semicircle Theorem An angle inscribed in a semicircle is a right angle. Theorem 5.3 Inscribed Angles in the Same Arc Theorem

6 Two or more angles inscribed in the same arc are congruent. Given: m GOF = 7x + 10, m GFO = 8x + 20, m DOF = 3y 12, m LFO = 2y 3, and OL DF Find: m GOF, m DOF, and m LFO Prove this: Given: O with mm A = 45 Prove: PO YO Let's Practice: Direction: Use the given figures to find the value of x

7 Lesson 5.3: Tangents Definition 5.13 Tangent to a Circle A line in the plane of the circle that intersects the circle at exactly one point is called tangent line. Definition 5.14 Point of Tangency It is the point of intersection between a tangent line and a circle. Note: A circle separates a plane into three parts: 1. the interior; 2. the exterior; and 3. the circle itself. Theorem 5.4 The Tangent-Line Theorem If a line is tangent to a circle, then it is perpendicular to the radius at its outer endpoint. AP is tangent to C at P. If CP is 7cm and AP is 24cm, how far is point A from center C? Theorem 5.5 The Converse of the Tangent-Line Theorem In a plane, if a line is perpendicular to a radius of a circle at the endpoint on the circle, then the line is a tangent to the circle. Note: Theorem 5.5 can be used to identify tangents to a circle. In the figures ES is 8cm and PS is 15cm. Show that PS is tangent to E.

8 Theorem 5.6 The Tangent-Segment Theorem If two tangent segments are drawn to a circle from an external point, then: (a) the two tangent segments are congruent, and (b) the angles between the tangent segments and the line joining the external point to the center of the circle are congruent. Given: OT and OU are tangent segments to C Prove: (a) OT OU and (b) TOC UOC Definition 5.14 Common Tangent A line or a segment that is tangent to two circles in the same plane is called a common tangent of the two circles. Two Types of Common Tangent 1. Common External Tangent Common external tangents do not intersect the segment whose endpoints are the centers of the circles. Figure Common Interior Tangents Common interior tangents intersect the segments whose endpoints are the centers of the circles. Figure 5.5 Theorem 5.7 The Tangent Circles Theorem If two circles are tangent internally or externally, then their line of centers pass through the point of contact.

9 Figure 5.6 Internally Tangent Circles Figure 5.7 Externally Tangent Circles Given: line m is tangent to A and at P Prove: AB passes through P Definition 5.15 Tangent Circles These are two circles whose intersection is exactly one point. Definition 5.16 Line of Centers It is the segment joining the centers of two circles. Definition 5.17 Common Tangent It is a line which is tangent to two circles. Definition 5.18 Common Internal Tangent It is a common tangent which intersects the line of centers. Definition 5.19 Common External Tangent It is a common tangent which does not intersect the line of centers. Definition 5.20 Internally Tangent Circles These are tangent circles whose common tangent does not intersect the line of centers. Definition 5.21 Externally Tangent Circles These are tangent circles whose common tangent intersects the line of centers. Prove this: Given: BS and ES are tangent to D Prove: (a) BS ES and (b) BSD ESD

10 Let's Practice: Direction: Refer to the figure at the right. AM and CH are common external tangents of R and I. MY and DT are common internal tangents of R and I. If QI is 10, RE is 4, and m TIQ is 60, find the measures of the following: 1. m ITQ = 2. m TQI = 3. TS = 4. SI = 5. IY = 6. RD =

11 Lesson 5.4: Chords and Arcs Theorem 5.8 The perpendicular from the center of the circle to any chord bisects the chord. Given: OB HT Prove: OB bisects HT Theorem 5.9 The line joining the center of the circle to the midpoint of any chord which is not a diameter is perpendicular to the chord. Given: O with LV a chord and E the midpoint of LV Prove: OE LV Theorem 5.10 The perpendicular bisector of a chord of a circle passes through the center of the circle. Given: O with AB the perpendicular bisector of KT Prove: AB passes through O Theorem 5.11 The perpendicular bisector of a chord of a circle bisects the central angle subtended by the chord. Given: O with OT a perpendicular bisector of AM Prove: OT bisects AOM

12 Theorem 5.12 The bisector of a central angle subtended by the chord is the perpendicular bisector of the chord. Given: O with OT an angle bisector of AOM Prove: OT is a perpendicular bisector of AM Theorem 5.13 In the same circle or in congruent circles, chords are congruent if and only if their distances from the center(s) of the circle(s) are equal. Note: Congruent circles are those whose radii are congruent. Given: Q with QR QS, QR AB and QS CD Prove: AB CD Theorem 5.14 In a circle or in congruent circles, two minor arcs are congruent if and only if their corresponding chords are congruent. In T, AS = 20cm and TS is 24 cm from the center of the circle. Find the radius of the circle. Let's Practice: Direction: Find x in each figure

13 Lesson 5.5: Angles Formed by Secants, Tangents, and Chords There are four ways for the two intersecting lines to intersect a circle. These are: (1) inscribed angle; Figure 5.7 (2) an angle formed by a tangent and a secant; Figure 5.8 (3) an angle formed by two secants intersecting in the interior of the circle; and Figure 5.9 (4) an angle formed by two secants intersecting in the exterior of the circle. Figure 5.10

14 Theorem 5.15 The Intersecting Secants-Exterior Theorem The measure of an angle formed by two secants that intersect in the exterior of a circle is one-half the difference of its intercepted arcs. Given: O, with secants AB and AC Prove: m A = 1 (mbc mde ) 2 The angle between two secants intersecting in the exterior of the circle is 55. If one of the intercepted arcs measures 150, what is the degree measure of the other arc? Theorem 5.16 The measure of an angle formed by a tangent and a secant drawn at the point of contact is one-half the measure of its intercepted arc. Given: O with BC at point B Prove: m ABC = 1 2 mab as tangent and intersecting secant AB In the given circle, HK is a diameter with tangent ray HW. Find m WHI. Theorem 5.17 The Intersecting Secants-Interior Theorem The measure of an angle formed by two secants intersecting in the interior of the circle is equal to one-half the sum of the measures of its intercepted arcs.

15 Given: AB intersects CD at P Prove: m DPB = 1 (mbd + mac ) 2 Refer to the circle at the right. (a) If mbn is 60 and mar is 70, find m BDN. (b) If mnba is 260 and mbr is 80, find m ADN. Let's Practice: Direction: Using theorems we previously studied, find each measure if OA M at A and L, respectively, while mul is 120, mla is 90, and mca is muc = 2. m LCA = 3. m LUA = 4. m B 5. m UNL = 6. m LNA = 7. m CAU = 8. m CAO = 9. m UAS = 10. m ASL = 11. m CLS = 12. m CLT = 13. m BLT = 14. m UNC = 15. mcul = Direction: Find the value of x. and LT are tangents to

16 Lesson 5.6: The Power Theorems Theorem 5.18 The Intersecting Segments of Chords Power Theorem If two chords intersect in the interior of the circle, then 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. Two chords EK and IS intersect at P as shown in the figure at the right. a. If the segments of EK have lengths 9cm and 18cm, find the length of the other segment of the chord if the length of one of the segments of the chord is 27 cm. b. If EP = 3, PK = 18, IP = 3x, and PS = 2x, find x. Theorem 5.19 The Segments of Secants Power Theorem If two secants intersect in the exterior of the circle, the product of the length of one secant segment and the length of its external part is equal to the product of the length of the other secant segment and the length of its external part. Use the figure at the right to answer each of the following: a. If SL = 12, DL = 18, and SU = 16, find SB. b. If SL = 6, DL = x, SU = 8, and BU = 18, find DL. Theorem 5.20 The Tangent Secant Segments Power Theorem If a tangent segment and a secant intersect in the exterior of a circle, then the square of the length of the tangent segment is equal to the product of the lengths of the secant segment and its external part. In the figure BS is tangent to T at S. a. If BC = 8cm, and CU = 10cm, find BS. b. If BS = 6cm, BC = x, and CU = x + 1, find BU.

17 Let s Practice: Direction: Find the value of x