c.) If RN = 2, find RP. N

Transcription

1 Geometry 10-1 ircles and ircumference. Parts of ircles 1. circle is the locus of all points equidistant from a given point called the center of the circle.. circle is usually named by its point. 3. The circle below is called circle K, written d K. ny segment with endpoints that are on the circle is a chord of the circle. and are chords. chord that passes through the of the circle is a diameter of the circle. is a diameter of the circle. K F ny segment with endpoints that are the and a point on the circle is a radius. KF, K, and K are radii of the circle. x 1: Identify parts of the circle. a.) Name the circle. b.) Name a radius of the circle. c.) Name a chord of the circle. 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. diameter is composed of two radii, so, d = r Q x : ircle R has diameters ST and QM. a.) If ST = 18, find RS. b.) If RM = 4, find QM. S P R T c.) If RN =, find RP. N M

2 x 3: The diameters of d X, d Y, and d Z are millimeters, 16 millimeters, and 10 millimeters respectively. a.) Find Z. b.) Find XF. X Y F Z G. ircumference. 1. The circumference is the distance around a circle.. The circumference around a circle is often represented by the letter. 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 For a circumference of units, diameter of d, or radius of r, = π d or = rπ x 4: Find the missing value for each circle. a.) Find if r = 13 inches. b.) Find if d = 6 millimeters. c.) Find d and r to the nearest x 5: Find the exact circumference for d K. 3 K HW: Geometry 10-1 p odd, 58-60, 63-64, Hon: 61, 65, 73-74

3 Geometry 10- ngles and rcs. ngles and rcs. 1. central angle has the of the circle as the vertex and its sides contain two radii of the circle.. Sum of entral ngles - the sum of the measures of a circle with no interior points in common is 360. m 1+ m + m 3 = 360 x 1: Refer to d T. a.) Find m RTS. b.) Find m QTR. Q T R (8x 4) o (13x 3) o 0x o (5x + 5) o 1 3 S U V example minor arc major arc semicircle» 110o ¼ J K F 40 o G ¼JKL F ¼JML L M named: arc degree measure = by the two letters of the endpoints» the measure of the central angle is less than 180. o m = 110 so m» = 110 by the letters of two endpoints and another point on the arc ¼ F greater than 180, but less than 360. mf ¼ = 360 m» mf ¼ = by the letters of the two endpoints and another point on the arc JKL ¼ and JML ¼ mjkl ¼ = mjml ¼ = 3. Theorem In the same or in congruent circles, two arcs are congruent if and only if their corresponding central angles are congruent.

4 P 4. Postulate 10-1 rc ddition Postulate - The measure of an arc formed by two adjacent arcs is the sum of the measures of the two arcs. In d S, mpq» + mrq» = mpqr ¼. S R Q x 3: In d P, m NPM = 46, PL bisects KPM, and OP KN. Find each measure. L a.)» mok b.) c.) ¼ mlm ¼ mjko J K O P N M x 4: This graph shows the percent of each type of bicycle sold in the United States in 001. icycles ought In 001 (by type) a.) Find the measurement of the central angle representing each category. Youth - 6%(360) = Mountain - = omfort - Road - Hybrid - b.) Is the arc for the wedge named Youth congruent to the arc for the combined wedges named Road and omfort? Youth 6% Hybrid 9% Road 7% Mountain 37% omfort 1%. rc Length 1. n arc is part of a circle, so arc length is part of the circumference. x 5: In length of» d, = 9, and m = 40, Find the HW Geometry 10- p odd, 47-51, 53, 57-6, 66, Hon: 63, 67

5 Geometry 10-3 rcs and hords. rcs and hords 1. Theorem In a circle or in congruent circles, two minor arcs are congruent if and only if their corresponding chords are congruent. bb. - In d, minor arcs are, corr. chords are. - In d, chords are, corr. minor arcs are. x: If,»», and if»», x 1: Finish the proof of Theorem 10- U Given: d X, UV» YW» V Prove: UV YW X W Statements Reasons 1. d X, UV» YW» 1. Given Y. UXV WXY. If arcs are 3. UX VX WX YX UV YW 5.. iameters and hords 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. x: If TV, then and»». T U V M x : ircle W has a radius of 10 cm. Radius WL is perpendicular to chord HK, which is 16 cm. long. H W J K L

6 a.) If mhl» = 53, find mmk ¼. b.) Find JL.. Theorem 10-4 In a circle or in congruent circles, two chords are congruent if and only if they are equidistant from the center. x 3: hords F and GH are equidistant from the center. If the radius of d P is 15 and F = 4, find PR and RH. Q P F H R G HW: Geometry odd, 39-41, 48-49, 5-65 Hon: 36, 4-43

7 Geometry 10-4 Inscribed ngles. Inscribed angles 1. n inscribed angle is an angle that has its vertex and its sides contained in the chords of the circle. Vertex is on the circle» is the arc intercepted by and are chords on the circle. Theorem Inscribed ngle 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 arc is twice the measure of the inscribed angle). x: m = 1 ( m» ) or ( m ) = m» x 1: In d F, mwx ¼ = 0, mxy» = 40, muz» = 108, and muw ¼ = myz». Find the measures of the numbered angles. W X Y F U 1 T Z 3. Theorem If two inscribed angles of a circle (or congruent circles) intercept congruent arcs or the same arc, then the angles are congruent. bbreviations: Inscribed s of arcs are. Inscribed s of same arc are.

8 P Q K x : Given: d with QR GF, and JK HG Prove: VPJK V HG J H F R Statements Reasons 1. QR GF, JK HG 1.. QR» GF». G 3. GF intercepts FG» 3. QPR intercepts QR» VPJK V HG 6.. ngles of Inscribed Polygons 1. Theorem If an inscribed angle intercepts a semicircle, the angle is a right angle. ¼ is a semicircle so m = 90 o x 3: Triangles TVU and TSU are inscribed in d P with VU» SU». Find the measure of each numbered angle if m = x + 9 and m 4 = x + 6. V 3 4 U 1 T S x 4: Quadrilateral QRST is inscribed in d M. If m Q = 87 and m R = 10. Find m S and m T.. Theorem If a quadrilateral is inscribed in a circle, then its opposite angles are supplementary. HW: Geometry 10-4 p , 13-16, 18-0, -9, 40-43, Hon: 11, 17, 31-3, 35

9 Geometry 10-5 Tangents. Tangents 1. is tangent to d, because the line containing uuur intersects the circle in exactly one point.. This point is called the point of tangency. 3. Theorem If a line is tangent to a circle, then it is perpendicular to the radius drawn to the point of tangency. K suur If is a tangent, x 1: RS is tangent to K d Q at point R. Find the diameter.. 0 S 16 P Q R 4. Theorem 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. x : a.) etermine whether b.) etermine whether is tangent to d. W is tangent to d W 5. Theorem If two segments from the same exterior point are tangent to a circle, they are congruent.

10 x 3: Find x, assume that segments that appear to be tangent are. G T H y - 5 S F x y. ircumscribed 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.) x 4: Find the perimeter of the triangle. Given: = 6 19 F HW: Geometry 10-5 p , 3-5, 9-30, 33-40, 4-45 Hon: 0, 6

11 Geometry 10-6 Secants, Tangents, and ngle Measures. Intersections on or inside a circle. 1. line that intersects a circle in exactly two points is called a. - secant of a circle contains a chord of the circle.. Theorem 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» + m» ) xamples: m 1 = 1 ( m» + m» ) m = F 88 x 1: Find m 4 if mfg» = 88 and mh ¼ = H G 3. Theorem 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. 180 o x : Find m RPS if mpt» = 114 o and mts º = 136 o P R Q S 114 o 136 o T

12 . Intersections outside of a circle. 1. Theorem 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 1 m = ( m» m» ) m = 1 ( m» m» ) 1 m = ( m ¼ m» ) x 3: Find x. 141 o xo 6 o x 4: 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. 40 o x 5: Find x. 55 o 40 o 6x o HW: Geometry 10-6 p odd, 34-39, 44, 47-55, Hon: 8, 3, 40a

13 Geometry 10-7 Special Segments in a ircle. Segments intersecting in a circle. 1. Theorem If two chords intersect in a circle, then the products of the measures of the segments of the chords are equal. So =. x 1: Find x. 1 x 8 9 x : iologists often examine organisms under microscopes. The circle represents the field of view under the microscope with a diameter of mm. etermine the length of the organism is it is located 0.5 mm from the bottom of the field of view. Round to the nearest hundredth of a millimeter. 0.5 mm. Segments intersecting outside a ircle 1. Theorem 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 and its external secant segment. = x 3: Find HJ if F = 10, H = 8, and FG = 4. H F J G

14 . 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 W X secant segment. ( WX ) = WZ WY Y Z x 4: Find x, assume that segments that appear to be tangent are. 3 6 x x 5: Find x, assume that segments that appear to be tangent are. x + x x + 4 HW: Geometry 10-7 p , 3, 5-9, 34-41, 43-48

15 Geometry 10-8 quations of ircles. quation of a ircle 1. Suppose the center of a circle is at (3, ), and the radius is 4.. Use the distance formula to determine the distance to a point on the circle. (3, ) P(x, y) d = ( x x ) + ( y y ) = ( x ) + ( y ) 16 = ( ) + ( ) this is the general form for an equation of a circle. 3. n equation for a circle with center at (h, k) and radius of r units, is ( x h) + ( y k) = r x 1: Write an equation for each circle. a.) center at (4, -3), r = 6 b.) center at (-1, -1), d = 16. x : 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.

16 x 3: a.) Graph ( x ) + ( y + 3) = 4 c.) Graph ( x 3) y 16 + = HW: Geometry 10-8 p odd, 35-37, Hon: 38-39, 54-55