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Geometry 10-1 Circles and Circumference A. Parts of Circles 1. A circle is the locus of all points equidistant from a given point called the center of the circle. 2. A circle is usually named by its ________ point. 3. The circle below is called circle K, written d K . A C Any segment with endpoints that are on the circle is a chord of the circle. AB and CD are chords. Any segment with endpoints that are the _______ and a point on the circle is a radius. KF , KC , and KD are radii of the circle. K B A chord that passes through the ________ of the circle is a diameter of the circle. CD is a diameter of the circle. F D Ex 1: Identify parts of the circle. a.) Name the circle. B A E b.) Name a radius of the circle. C c.) Name a chord of the circle. D d.) Name a diameter of the circle. 4. The distance from the center of the circle to any point on the circle is always the same, therefore, all radii are congruent. 5. A diameter is composed of two radii, so, d = 2r P Ex 2: Circle R has diameters ST and QM . a.) If ST = 18, find RS. b.) If RM = 24, find QM. c.) If RN = 2, find RP. S Q R N M T Ex 3: The diameters of d X , d Y , and d Z are 22 millimeters, 16 millimeters, and 10 millimeters respectively. a.) Find EZ. E X Y F Z G b.) Find XF. B. Circumference. 1. The circumference is the distance around a circle. 2. The circumference around a circle is often represented by the letter C. 3. The ratio between the circumference of a circle and it’s diameter is about 3:1. For each diameter, circumference is a little more than 3. 4. For a circumference of C units, diameter of d, or radius of r, C = π d or C = 2rπ Ex 4: Find the missing value for each circle. a.) Find C if r = 13 inches. b.) Find C if d = 6 millimeters. c.) Find d and r to the nearest Ex 5: Find the exact circumference for d K . 3 2 K HW: Geometry 10-1 p. 526-528 17-55 odd, 58-60, 63-64, 75-80 Hon: 61, 65, 73-74 Geometry 10-2 Angles and Arcs A. Angles and Arcs. 1. A central angle has the __________ of the circle as the vertex and its sides contain two radii of the circle. 2. Sum of Central Angles - the sum of the measures of a circle with no interior points in 1 3 common is 360. 2 m∠1 + m∠2 + m∠3 = 360 R Ex 1: Refer to d T . a.) Find m∠RTS . Q S (8 x − 4)o T (13x − 3)o 20xo b.) Find m∠QTR . (5 x + 5)o U V example minor arc A B major arc »AC 110 o C named: by the two letters of the endpoints »AC arc degree measure = the measure of the central angle is less than 180. m∠ABC = 110o so m »AC = 110 D E 40 o semicircle ¼ DFE G F J ¼ JML M K ¼ JKL L by the letters of two by the letters of the endpoints and another two endpoints and ¼ another point on the point on the arc DFE ¼ and JML ¼ arc JKL ¼ = ____ greater than 180, but mJKL less than 360. ¼ = ____ mJML ¼ = 360 − mDE » mDFE ¼ = ___ mDFE 3. Theorem 10-1 - In the same or in congruent circles, two arcs are congruent if and only if their corresponding central angles are congruent. P 4. Postulate 10-1 Arc Addition Postulate - The measure of an arc formed by two adjacent arcs is the sum of the measures of the two arcs. » + mRQ » = mPQR ¼ . In d S , mPQ S Q R Ex 3: In d P , m∠NPM = 46 , PL bisects ∠KPM , and OP ⊥ KN . Find L each measure. K » a.) mOK ¼ b.) mLM M P J ¼ c.) mJKO N O Ex 4: This graph shows the percent of each type of bicycle sold in the United States in 2001. a.) Find the measurement of the central angle representing each category. Youth - 26%(360) = _____ Mountain - ____________ = ____ Comfort Road Hybrid - Bicycles Bought In 2001 (by type) Hybrid 9% b.) Is the arc for the wedge named Youth congruent to the arc for the combined wedges named Road and Comfort? B. Arc Length 1. An arc is part of a circle, so arc length is part of the circumference. Ex 5: In d B , AC = 9, and m∠ABD = 40 , Find the length of »AD HW Geometry 10-2 p. 533-535 15-43 odd, 47-51, 53, 57-62, 66, 69-76 Hon: 63, 67 Mountain 37% Youth 26% Comfort 21% Road 7% A B D C Geometry 10-3 Arcs and Chords A. Arcs and Chords 1. Theorem 10-2 - In a circle or in congruent circles, two minor arcs are congruent if and only if their corresponding chords are congruent. Abb. - In d , 2 minor arcs are ≅ , corr. chords are ≅ . - In d , 2 chords are ≅ , corr. minor arcs are ≅ . A D » , and if »AB ≅ CD » , AB ≅ CD Ex: If AB ≅ CD, »AB ≅ CD Ex 1: Finish the proof of Theorem 10-2 C U V » ≅ YW » Given: d X , UV X Prove: UV ≅ YW Statements » ≅ YW » 1. d X , UV 2. ∠UXV ≅ ∠WXY B W Reasons Y 1. Given 2. If arcs are ____________________ _______________________________ 3. UX ≅ VX ≅ WX ≅ YX 3.______________________________ 4. __________________ 4.____________ 5. UV ≅ YW 5.________________ B. Diameters and Chords 1. Theorem 10-3 In a circle, if a diameter (or radius) is perpendicular to a chord, then it bisects the chord and its arc. Ex: If BA ⊥ TV , then _______ and » ≅ » . B T C U V A M Ex 2: Circle W has a radius of 10 cm. Radius WL is perpendicular to chord HK , which is 16 cm. long. H W J L K » = 53 , find mMK ¼ . a.) If mHL b.) Find JL. 2. Theorem 10-4 In a circle or in congruent circles, two chords are congruent if and only if they are equidistant from the center. Ex 3: Chords EF and GH are equidistant from the center. If the radius of d P is 15 and EF = 24 , find PR and RH. Q E P G HW: Geometry 540-543 11-35 odd, 39-41, 48-49, 52-65 Hon: 36, 42-43 F H R Geometry 10-4 Inscribed Angles A. Inscribed angles 1. An inscribed angle is an angle that has its vertex and its sides contained in the chords of the circle. B Vertex B is on the circle »AC is the arc intercepted by ∠ABC AB and BC are chords on the circle A C 2. Theorem 10-5 - Inscribed Angle Theorem - If an angle is inscribed in a circle, then the measure of the angle equals one-half the measure of its intercepted arc (or the measure of the intercepted B arc is twice the measure of the inscribed angle). 1 » » Ex: m∠ABC = (m AC ) or 2(m∠ABC ) = m AC 2 A C ¼ = 20 , m »XY = 40 , mUZ » = 108 , and mUW ¼ = mYZ » . Ex 1: In d F , mWX W X Find the measures of the numbered angles. Y 3 4 5 F 2 U 1 Z T 3. Theorem 10-6 - If two inscribed angles of a circle (or congruent circles) intercept congruent arcs or the same arc, then the angles are congruent. B A Abbreviations: Inscribed ∠ ’s of ≅ arcs are ≅ . Inscribed ∠ ’s of same arc are ≅ . ∠DAC ≅ ∠ ____ D C Q P K Ex 2: Given: d C with QR ≅ GF , C and JK ≅ HG Prove: VPJK ≅VEHG Reasons 1. » 3. ∠GEF intercepts FG » ∠QPR intercepts QR 4. 5. 6. VPJK ≅VEHG 3. R F H E Statements 1. QR ≅ GF , JK ≅ HG » ≅ GF » 2. QR J G 2. 4. 5. 6. B. Angles of Inscribed Polygons 1. Theorem 10-7 - If an inscribed angle intercepts a semicircle, the angle is a right angle. A D ¼ ADC is a semicircle so m∠ABC = 90o C B Ex 3: Triangles TVU and TSU are inscribed in » ≅ SU » . Find the measure of each d P with VU numbered angle if m∠2 = x + 9 and m∠4 = 2 x + 6 . V 3 1 T Ex 4: Quadrilateral QRST is inscribed in d M . If m∠Q = 87 and m∠R = 102 . Find m∠S and m∠T . 2. Theorem 10-8 - If a quadrilateral is inscribed in a circle, then its opposite angles are supplementary. HW: Geometry 10-4 p. 549-551 8-10, 13-16, 18-20, 22-29, 40-43, 46-58 Hon: 11, 17, 31-32, 35 U 4 2 S Geometry 10-5 Tangents A. Tangents 1. BC is tangent to d A , because the line uuu r containing BC intersects the circle in exactly one point. 2. This point is called the point of tangency. A B C 3. Theorem 10-9 - If a line is tangent to a circle, then it is perpendicular to the radius drawn to the point of tangency. K suu r If AB is a tangent, AB ⊥ AK A B S Ex 1: RS is tangent to d Q at point R. Find the diameter.. 20 P 16 R Q 4. Theorem 10-10 - In a plane, if a line is ______________ to a radius of a circle at the endpoint on the circle, then the line is a tangent to the circle. Ex 2: a.) Determine whether BC is tangent to d A . b.) Determine whether WE is tangent to d D . E 16 A 7 D 7 10 7 B 9 10 C 5. Theorem 10-11 - If two segments from the same exterior point are tangent to a circle, they are congruent. 24 W A B AB ≅ BC C Ex 3: Find x, assume that segments that appear to be tangent are. T H G y-5 x+4 S E F 10 y B. Circumscribed Polygons 1. Polygons can be circumscribed about a circle, or the circle is inscribed in the polygon. (every side of the polygon is tangent to the circle.) Ex 4: Find the perimeter of the triangle. Given: AD = CB D C 6 E 19 A HW: Geometry 10-5 p. 556-558 8-19, 23-25, 29-30, 33-40, 42-45 Hon: 20, 26 F B Geometry 10-6 Secants, Tangents, and Angle Measures A. Intersections on or inside a circle. 1. A line that intersects a circle in exactly two points is called a ______________. -A secant of a circle contains a chord of the circle. 2. Theorem 10-12 - If two secants intersect in the interior of a circle, then the measure of an angle formed is one-half the sum of the measures of the arcs intercepted by the angle and its vertical angle. » ) (m »AC + mDB Examples: m∠1 = 2 m∠2 = A 1 B C » ) (m »AD + mBC 2 F » = 88 and mEH ¼ = 76 Ex 1: Find m∠4 if mFG 88 G 4 3 E 76 3. Theorem 10-13 - If a secant and a tangent intersect at the point of tangency, then the measure of each angle formed is ____________ the measure of its intercepted arc. D 2 H 180o R » = 114o and mTS º = 136o Ex 2: Find m∠RPS if mPT P Q S 114o T 136o B. Intersections outside of a circle. 1. Theorem 10 -14 - If two secants, a secant and a tangent, or two tangents intersect in the exterior of a circle, then the measure of the angle formed is one-half the positive difference of the measures of the intercepted arcs. Two secants Secant - Tangent Two Tangents D D E D B B B C C A A C A 1 » 1 » » ) » ) − mBC − mBC m∠A = (mDE m∠A = (mDC 2 2 1 ¼ » ) − mBC m∠A = (mBDC 2 Ex 3: Find x. 141o xo 62o Ex 4: A jeweler wants to craft a pendant with the shape shown. Use the figure to determine the measure of the arc at the bottom of the pendant. Ex 5: Find x. 55o 40o HW: Geometry 10-6 p. 564-568 13-31odd, 34-39, 44, 47-55, 57-59 Hon: 28, 32, 40a 6x o 40o Geometry 10-7 Special Segments in a Circle A. Segments intersecting in a circle. 1. Theorem 10-15 - If two chords intersect in a circle, then the products of the measures of the segments of the chords are equal. A E So AE ⋅ EC = DE ⋅ EB . Ex 1: Find x. 12 8 B C D x 9 Ex 2: Biologists often examine organisms under microscopes. The circle represents the field of view under the microscope with a diameter of 2 mm. Determine the length of the organism is it is located 0.25 mm from the bottom of the field of view. Round to the nearest hundredth of a millimeter. 0.25 mm B. Segments intersecting outside a Circle 1. Theorem 10-16 - If two secant segments are drawn to a circle from an ___________ _________, the product of the measures of one secant segment and its external secant segment is equal the product of the measures of the other secant segment A B and its external secant segment. C D AB ⋅ AC = AD ⋅ AE Ex 3: Find HJ if EF = 10, EH = 8, and FG = 24. E E H J F G 2. If a _______________ segment and a _______________ segment are drawn to a circle from an exterior point, then the square of the measure of the tangent segment is equal to the product of the measures of the secant segment and its external X W secant segment. (WX ) 2 = WZ ⋅ WY Y Z Ex 4: Find x, assume that segments that appear to be tangent are. 6 3 Ex 5: Find x, assume that segments that appear to be tangent are. x+2 x x+4 HW: Geometry 10-7 p. 572-574 8-19, 23, 25-29, 34-41, 43-48 x Geometry 10-8 Equations of Circles A. Equation of a Circle 1. Suppose the center of a circle is at (3, 2), and the radius is 4. 2. Use the distance formula to determine the distance to a point on the circle. P(x, y) (3, 2) d = ( x2 − x1 ) 2 + ( y2 − y1 ) 2 4 = ( x − __) 2 + ( y − __) 2 16 = (_____) 2 + (_____) 2 this is the general form for an equation of a circle. 3. An equation for a circle with center at (h, k) and radius of r units, is ( x − h) 2 + ( y − k ) 2 = r 2 Ex 1: Write an equation for each circle. a.) center at (4, -3), r = 6 b.) center at (-12, -1), d = 16. Ex 2: A circle with a diameter of 10 has its center in the first quadrant. The lines y = −3 and x = −1 are tangent to the circle. Write an equation of the circle. Ex 3: a.) Graph ( x − 2) 2 + ( y + 3)2 = 4 c.) Graph ( x − 3) 2 + y 2 = 16 HW: Geometry 10-8 p. 578-580 11-33odd, 35-37, 47-53 Hon: 38-39, 54-55