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M2 GEOMETRY PACKET 1 FOR UNIT 7 – PROPERTIES OF CIRCLES ASSIGNMENT SHEET FOR PACKET 1 OF UNIT 7 This packet includes sections 10-1 TO 10-5 from our textbook and review for those sections. Date Due Number 7A 7B Assignment Topics p. 701 – 702 # 5, 6, 10-16 all, 24, 28, 32 p. 711 – 712 # 16-21 all, 36, 40, 45, 46, 47 7C p. 719 # 7-10 all, 16-19 all 7D p. 727 # 5, 6, 8-13 all 7E p. 736 – 737 # 4, 5, 6, 8, 14, 15, 24-27 all 10-1 Circles & Circumference (p. 2-4) Vocabulary: circle, center, radius, chord, diameter, concentric circles, circumference, pi, inscribed, circumscribed Name and identify parts of a circle. Solve problems involving circumference of a circle. 10-2 Measuring Angles & Arcs (p. 5-7) Vocabulary: central angle, arc, minor arc, major arc, semicircle, congruent arcs, adjacent arcs, arc length Use algebra to solve problems involving arcs of a circle. 10-3 Arcs & Chords (p. 8-9) Use algebra to solve problems involving chords. 10-4 Inscribed Angles (p. 10-12) Vocabulary: inscribed angle, intercepted arc Use algebra to solve problems involving inscribed angles. 10-5 Tangents (p. 13-14) Vocabulary: tangent, point of tangency, common tangent Use algebra to solve problems involving tangents to a circle. Quiz on 10-1 to 10-5 For all problems in this packet, label all answers with appropriate units, when given. 1 M2 GEOMETRY PACKET 1 FOR UNIT 7 – PROPERTIES OF CIRCLES SECTION 10-1 CIRCLES & CIRCUMFERENCE Circle: all points in a plane that are the same distance (radius) from a given point (center) Radius: (plural: radii) a segment with endpoints at the center of a circle and on the circle Chord: a segment with both endpoints on the circle Diameter: a chord that passes through the center of a circle Name circles using a circle symbol and a single capital letter (the center). For example, a circle with center O would be named O . Name radii, chords, and diameters as you would name any other segments, using 2 capital letters with a bar over the top. Use the circle with center R shown below to answer questions 1 through 7: 1. Name the circle. 2. Name all radii shown in the diagram. 3. Name all chords shown. 4. Name all diameters shown. 5. If AB = 18 mm, find AR. 6. If RY = 10 in., find AR and AB. 7. Is AB XY ? Explain. 2 M2 GEOMETRY PACKET 1 FOR UNIT 7 – PROPERTIES OF CIRCLES Recall that circumference is the distance around a circle. C d or C 2 r 8. Find the diameter and radius of the circle with given circumference. Round answers to the nearest hundredth. a. C = 40 in. b. C = 15.62 m 9. Use your answers above to find the ratio a. Use (a) above. c. C = 79.5 yd circumference . Round to the nearest hundredth. diameter b. Use (b) above. c. Use (c) above. 10. (a) Similar figures have the same shape and lengths that are in the same proportion. Are the circles in #8-9 similar to each other? Explain. (b) Are all circles similar to each other? Explain. To find exact circumference, leave as a symbol instead of typing it in your calculator. 11. Find the exact circumference of each circle. You may need to use the Pythagorean Theorem or other geometric rules to find the diameter first. a. b. c. 3 M2 GEOMETRY PACKET 1 FOR UNIT 7 – PROPERTIES OF CIRCLES 12. Fill in the blanks. The words in parts (d) and (g) have not come up yet. You’ll need to Google the definitions to find the appropriate terms. You can also check against the vocabulary list on page 1 of this packet. a. __________________________ is the distance around a circle. b. A segment with one endpoint at the center of a circle and the other on the circle is known as a ____________. c. A ____________ is the set of all points in a plane that are equidistant from a given point. d. ____________________ circles are coplanar and have the same center. (Google this one. It hasn’t come up in any problems yet.) e. A ________________ is chord that passes through the center of the circle. f. ____ is an irrational number that is found by the ratio of the circumference to the diameter of any circle. g. A polygon drawn inside a circle so that all of its vertices lie in the circle is said to be __________________ in the circle. That circle is said to be __________________________ around the polygon. (Google these.) h. A __________ is any segment with both endpoints on the circle. 4 M2 GEOMETRY PACKET 1 FOR UNIT 7 – PROPERTIES OF CIRCLES SECTION 10-2 MEASURING ANGLES & ARCS Part 1: Vocabulary and Naming 1. A central angle is an angle with the center of the circle as its vertex and two radii of the circle as its sides. Name 3 different central angles shown in E at right. Use the angle symbol in front of each name. ________ ________ ________ 2. The diagram of E shows 4 non-overlapping central angles. What is the sum of the measures of these 4 angles? An arc is a portion of the circumference of a circle. A central angle Here are some different types of arcs: A minor arc is the shortest arc between two points on a circle. Name a minor arc using its endpoints in any order with a curved arc over both letters. For example, is the shortest arc formed by CEH in E . This arc can also be named . 3. Give 2 different names for the minor arc formed by HEF . A major arc is the longest arc between two points on a circle. The name of a major arc must contain 3 letters, the endpoints with any other point on the arc between them. For example, the major arc formed by CEH in E can be named , or . 4. Give as many names as you can for the major arc formed by HEF in E at right, and emphasize this arc by making it bolder in the diagram. A semicircle is an arc with endpoints on a diameter of a circle. 5 M2 GEOMETRY PACKET 1 FOR UNIT 7 – PROPERTIES OF CIRCLES Part 2: Arc Measure Arcs have both length and measure. The measure of a minor arc is expressed in degrees and is equal to the measure of its central angle. The measure of a major arc is found by calculating 360 minus the measure of its minor arc. 5. (a) What is the measure of any semicircle? (b) The measure of any minor arc is between _______ and _______ . (c) Can the measure of a major arc be less than 180 ? Explain. 6. In R at right, AC is a diameter, and mARB 42 . Find the measure of each arc: ______ ______ ______ ______ ______ Part 3: Arc Length The length of an arc can be found using the following proportion: Part to whole for distances around the circle arc length central angle circumference 360 Part to whole for angles in the circle 7. Write a proportion, and solve to find the missing measure in O at right. Round to the nearest hundredth. a. Length of b. Length of if r = 2 m. if d = 7 in. 6 M2 GEOMETRY PACKET 1 FOR UNIT 7 – PROPERTIES OF CIRCLES arc length central angle circumference 360 8. Follow the steps below to solve this problem: The arc length between the blades of the wind turbine shown at right was calculated to be 309.8 feet. To the nearest whole number, what is the length of each blade of the turbine? a. Assuming the blades are set at equal distances around the circle, what is the degree measure of the angle between two blades? (This is the central angle.) b. Use the formula at the top of this page to write a proportion using the arc length given at the beginning of this problem, the central angle you found in part (a), and C as the circumference. c. Solve your proportion for C. d. Substitute the value of C above into the circumference formula: C 2 r . Then divide to solve for r, and answer the problem. 9. Follow the steps below to solve this problem: A stage is in the shape of a semicircle with radius 18 feet and the curved edge facing the audience. If a banner is hung all around the front of the state, how long will the banner need to be? Round to the nearest tenth. a. Calculate the exact circumference of a circle with radius 18 ft. b. What is the degree measure for the central angle of a semicircle? c. Use the formula at the top of this page to write a proportion using the circumference you found in part (a), the central angle you found in part (b), and x as the arc length. d. Answer the problem. 7 M2 GEOMETRY PACKET 1 FOR UNIT 7 – PROPERTIES OF CIRCLES SECTION 10-3 ARCS & CHORDS Part 1: Congruent Chords in a Circle 1. In E at right, chords AB and CD are congruent. a. Draw AEB and CED in the diagram. b. Since AEB CED by the ______ triangle congruence postulate, AEB ______. This shows that central angles formed by congruent chords in the same circle are __________________. Also, arcs formed by congruent chords are __________________. 2. In the same diagram above, draw a segment from E, perpendicular to AB , and another segment from E, perpendicular to CD . a. Explain why these 2 new segments must be congruent. b. Congruent chords in the same circle are ______________________ from the center. Conversely, if chords are ______________________ from the center of a circle, then they are congruent. 3. Find the value of x in each circle. a. b. c. d. CD CB , GQ x 5 , EQ 3 x 6 8 M2 GEOMETRY PACKET 1 FOR UNIT 7 – PROPERTIES OF CIRCLES Part 2: Perpendicular Bisector of a Chord 4. Open the applet at https://www.geogebra.org/o/SteKWsKP. 5. In the applet, drag point E until it is the midpoint of chord CD . If you have difficulty making this work, you can also move point C or D. 6. Drag point F until the measure of FED is approximately 90 . 7. The perpendicular bisector of a chord passes through the ____________ of the circle and is a ________________ of the circle. Conversely, if a ________________ of a circle is perpendicular to a chord, then it also ______________ the chord. 8. In O , CD OE , OD = 15, and CD = 24. a. OE is the __________________________ ________________ of CD . b. Find ED. c. Find OE. 9. In P , the radius is 13, and RS = 24. a. Find RT. b. Find PT. (HINT: Draw another radius of P to form a right triangle.) c. Find TQ. 9 M2 GEOMETRY PACKET 1 FOR UNIT 7 – PROPERTIES OF CIRCLES SECTION 10-4 INSCRIBED ANGLES Part 1: Inscribed Angles and Intercepted Arcs An angle is inscribed in a circle if its vertex lies on the circle and both of its sides contain chords of the circle. The intercepted arc is the minor arc cut off by the sides of the angle. 1. In each circle, name a central angle, an inscribed angle, and the intercepted arc formed by both angles. a. b. Central angle: ______ Central angle: ______ Inscribed angle: ______ Inscribed angle: ______ Intercepted arc: ______ Intercepted arc: ______ Part 2: The Measure of an Inscribed Angles and its Intercepted Arc 2. Open the applet at https://www.geogebra.org/o/Kh756zzu. 3. Drag point B, C, or D to change the angle measures, and use a calculator to compute the ratio shown on the screen, rounded to the nearest tenth. (Do not drag any of these points past each other around the circle.) Repeat until you’ve calculated the ratio at least 5 different times. 4. The measure of an inscribed angle is _________________ the measure of the central angle that intercepts the same arc. This means the measure of the inscribed angle is also _________________ the measure of its intercepted arc. Thus, if two inscribed angles intercept the same arc, the angles must be __________________. 10 M2 GEOMETRY PACKET 1 FOR UNIT 7 – PROPERTIES OF CIRCLES 5. Find the requested measure. a. ______ b. ∠ = ______ c. 6. Use the circle at right to answer the following: a. Which angle intercepts the same arc as ∠ ? b. Which angle intercepts the same arc as ∠ ? c. Find the measure of ∠ . d. Find the measure of ∠ . 7. In A at right, BC is a diameter. a. What is the measure of ? b. What is the measure of D ? c. An inscribed angle that intercepts a semicircle measures ________ . 11 ______ M2 GEOMETRY PACKET 1 FOR UNIT 7 – PROPERTIES OF CIRCLES Part 2: Angles of Inscribed Polygons In an inscribed polygon, all sides are chords, and all vertices are points on the circle. 8. In quadrilateral ABCD, diagonal BD is a diameter of R . a. mA ______ ; ______ b. mC ______ ; ______ c. mA mC ______ ; d. + ______ = ______ ; mABC mADC ______ e. If a quadrilateral is inscribed in a circle, then its opposite angles are _________________________. 9. Find the value of x in each. a. b. c. 12 M2 GEOMETRY PACKET 1 FOR UNIT 7 – PROPERTIES OF CIRCLES SECTION 10-5 TANGENTS A tangent is a line, ray, or segment that intersects a circle in exactly one point, called the point of tangency. A common tangent is tangent to two circles in the same plane. Part 1: Relationship Between Tangent and Radius 1. Open the applet at https://www.geogebra.org/m/e4qx3dFC. 2. Drag point C until ⃖ ⃗ appears to be tangent to the circle. 3. A line tangent to a circle is __________________________ to a radius at a point of tangency. 4. In each diagram below, assume that segments that appear to be tangent are tangent. Find the value of x in each: a. b. AD = 9, BC = 8 Part 2: Relationship Between Two Tangents 5. Open the applet at https://www.geogebra.org/m/ZwjDjfqG. 6. Right-click the pink and orange segments, and check Show Label to show the length of each segment. 7. Drag point A around the screen, observing the lengths of the pink and orange segments. 8. If two segments from the same point are tangent to a circle, then they are __________________. 9. Find x in each. Assume that segments that appear to be tangent are tangent. a. b. 13 M2 GEOMETRY PACKET 1 FOR UNIT 7 – PROPERTIES OF CIRCLES Part 3: Relationship Between Two Tangents When a polygon is circumscribed about a circle, all of the sides of the polygon are tangent to the circle. 10. Find x. Then find the perimeter of the polygon. a. b. c. d. 14