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Ch 6 Note Sheet L1 Shortened Key.doc Review: Circles Vocabulary Name ___________________________ If you are having problems recalling the vocabulary, look back at your notes for Lesson 1.7 and/or page 69 – 71 of your book. Also, pay close attention to the geometry notation you need to use to name the parts!! Circles G Circle is a set of points in a plane a given distance (radius) from a given point (center). F R Congruent circles are two or more circles with the same radius measure. Concentric circles are two or more circles with the same center point. A C P B D Types of Lines [Segments] Radius [of a circle] is A segment that goes from the center to any point on the circle. the distance from the center to any point on the circle. Diameter [of a circle] is A chord that goes through the center of a circle. the length of the diameter. d = 2r or ½ d = r. Chord is a segment connecting any two points on the circle. Secant A line that intersects a circle in two points. Tangent [to a circle] is a line that intersects a circle in only one point. The point of intersection is called the point of tangency. E radii: AP , PR , PB diameter: AB chords: AB , CD , AF secant: AG or AG tangent: EB or EB or BE … Arcs & Angles semicircles: ACB , ARB Arc [of a circle] is formed by two points on a circle and a continuous part of the circle between them. The two points are called endpoints. minor arcs: AC , AR , RD , BC ,… Semicircle is an arc whose endpoints are the endpoints of the diameter. Minor arc is an arc that is smaller than a semicircle. Major arc is an arc that is larger than a semicircle. Intercepted Arc An arc that lies in the interior of an angle with endpoints on the sides of the angle. Central angle An angle whose vertex lies on the center of a circle and whose sides are radii of the circle. Measure of a central angle determines the measure of an intercepted minor arc. S. Stirling major arcs: ARC , ARD , FRD , BAC ,… central angles with their intercepted arcs: mAPR mAR mRPB mRB Page 1 of 12 Ch 6 Note Sheet L1 Shortened Key.doc Lesson 6.1 Tangent Properties Investigation 1: See Worksheet page 1. Tangent Conjecture A tangent to a circle is perpendicular to the radius drawn to the point of tangency. Name ___________________________ Converse of the Tangent Conjecture A line that is perpendicular to a radius at its endpoint on the circle is tangent to the circle. O T N Investigation 2: See Worksheet page 1. Tangent Segments Conjecture Tangent segments to a circle from a point outside the circle are congruent. A E G Intersecting Tangents Conjecture The measure of an angle formed by two intersecting tangents to a circle is 180 minus the measure of the intercepted arc, x 180 a . x a a b “Tangent” means to intersect in one point. Tangent Circles Circles that are tangent to the same line at the same point. Internally Tangent Circles Two tangent circles having centers on the same side of their common tangent. Externally Tangent Circles Two tangent circles having centers on opposite sides of their common tangent. Polygons and Circles The triangle is inscribed in the circle, or the circle is circumscribed about the triangle. The triangle is circumscribed about the circle, or the circle is inscribed in the triangle. B All of the vertices of the polygon are on the circle. The sides are all chords of the circle. P C All of the sides of the polygon are tangent to the circle. O C A A S. Stirling B Page 2 of 12 Ch 6 Note Sheet L1 Shortened Key.doc Name ___________________________ Example 1 Example 2 Rays r as s are tangents. w = ? 70 5 cm 70 110 126 250 180 – 54 = w using formula OR Tangent radius, so mONM mOPM 90 Quad. sum = 360 so 360 180 110 x , x 70 y 5 cm Tangents to a circle from a point are congruent. mNP 110 Measure of a central angle equals its intercepted arc. central angle = w now use the kite formed by = tangent segments and = radii 360 – 90 – 90 – 54 = w 126 = w mPQN 360 110 250 Total degrees in a circle = 360 Example 3 AD is tangent to both circle B and circle C. w 100 mAXT 260 100 260 mA 90 and mD 90 Tangent radius, Quadrilateral sum = 360 so w 360 90 90 80 100 w mAT 100 Measure of central angle = intercepted arc and Circle’s degree = 360 so mAXT 360 100 260 S. Stirling Page 3 of 12 Ch 6 Note Sheet L1 Shortened Key.doc Lesson 6.2 Chord Properties Read top of page 317. Then use the diagrams to define the following. Central angle An angle whose vertex lies on the center of a circle and whose sides are radii of the circle. Measure of a central angle = measure of its intercepted arc. i.e. mBOA mAB Inscribed angle An angle whose vertex lies on a circle and whose sides are chords of the circle. Name ___________________________ Examples: Non-Examples: R D P O Q A S T B AOB , DOA and DOB are central PQR , PQS , RST , QST and QSR are NOT angles of circle O. central angles of circle O. Examples: Non-Examples: Q A C V R P B E T W X D U S ABC , BCD and CDE are inscribed PQR , STU , and VWX angles. are NOT inscribed angles. Investigation 3: See Worksheet page 4 top. Example 1 Chord Conjectures If two chords in a circle are congruent, then they determine two central angles that are congruent. their intercepted arcs are congruent. S. Stirling B D If AB CD , then O BOA COD . If AB CD , then A AB CD Page 4 of 12 C Ch 6 Note Sheet L1 Shortened Key.doc Lesson 6.3 Arcs and Angles Read top of page 324. Review (from Chapter 1.7): A circle measures 360. Name ___________________________ C Example 1 If mCOR 92 , then mCAR ? . 92 92 O 46 A R A semicircle measures 180. mCR 92 . The central angle equals its intercepted arc. Investigation 4: See Worksheet page 6. If mCR 92 , then mCAR 46 . The inscribed angle equals half its intercepted arc. Inscribed Angle Conjecture The measure of an angle inscribed in a circle equals half the measure of its intercepted arc. Example 2 mAB 170 , find mAPB and mAQB . 85 mAPB 1 170 85 2 mAQB 1 170 85 2 A P 85 Q 170 An inscribed angle measures ½ the intercepted arc. B Example 3 mAB mCD 42 , find mAPB and mCQD mAPB 1 42 21 2 1 mCQD 42 21 2 The inscribed angle equals half its intercepted arc. A P 42 21 B O C 42 21 Q D Example 4 ACD , ABD and AED intercept semicircle ARD . R A 180 Find the measure of each angle. All are 1 180 90 2 mACD mABD mAED 90 . S. Stirling C O 90 90 B Page 5 of 12 90 E D Ch 6 Note Sheet L1 Shortened Key.doc Name ___________________________ Investigation 5: See Worksheet page 6. Cyclic Quadrilateral A quadrilateral that can be inscribed in a circle. (Each of its angles are inscribed angles and each of its sides is a chord of the circle.) If ABDC is a cyclic quadrilateral mA mD 180 and mB mC 180 A Cyclic Quadrilateral Conjecture The opposite angles of a cyclic quadrilateral are supplementary. C mBAC = 106.61 mCDB = 73.39 mDBA = 85.99 B mACD = 94.01 D Cyclic Parallelogram Conjecture If a parallelogram is inscribed within a circle, then the parallelogram is a rectangle. G O Y D L Secant A line that intersects a circle in two points. Secants: EC , TN L Secant Segments: LN , SA , T SC , EA . Secant Segment A segment that intersects a circle in two points and it contains at least one point on the exterior of the circle. N Chords: EC , TN because they begin and end on the circle. S A C E Investigation 6: See Worksheet page 7. Parallel Lines [Secants] Intercepted Arcs Conjecture Parallel lines [secants] intercept congruent arcs on a circle. If AB DC , then mAD mBC 64 . D 64 A Note: mAB mDC . S. Stirling Page 6 of 12 C B 64 Ch 6 Note Sheet L1 Shortened Key.doc Name ___________________________ Are there any short cuts for finding the measures of angles and their intercepted arcs? Quick Formulas for Angles and Arcs: Diagram: Where is the What is the vertex of the figure formed angle? by? Center of the 2 radii A circle “Central O 1 angle” Measure of the angle formed? angle = intercepted arc m1 mAB B On the circle 2 chords “Inscribed angle” angle = ½ (intercepted arc) C E m2 1 mCE 2 2 D 1 chord & 1 tangent m2 1 mFG 2 F 2 H Inside the circle G 2 chords angle = ½ (sum of intercepted arcs) L m3 1 M 3 V 2 mJK mLM J K Outside the circle 2 secants angle = ½ (difference of intercepted arcs) N O 4 P 1 secant & 1 tangent Q W U T R 4 mNP mOQ m4 1 mWS mSU 2 m4 1 2 S 2 tangents m4 1 Z X 4 Y 2 mYAZ mYZ A Summary: Vertex at center → angle = arc Vertex on → angle = ½ arc Vertex inside → angle = ½ sum of arcs Vertex outside → angle = ½ difference between arcs S. Stirling Page 7 of 12 Ch 6 Note Sheet L1 Shortened Key.doc Name ___________________________ Lesson 6.5 The Circumference/Diameter Ratio Read top of page 335. Pi is a number, just like 3 is a number. It represents the number you get when you take the circumference of a circle and divide it by the diameter. There is no exact value for pi, so you use the symbol π. You will leave π in the answers to your problems unless they ask for an approximate answer. If they do, use 3.14 or the π key on your calculator. Circumference The perimeter of a circle, which is the distance around the circle. Also, the curved path of the circle itself. Circumference Conjecture If C is the circumference and d is the diameter of a circle, then there is a number π such that C = dπ Since d = 2r, where r is the radius, then C = 2rπ or 2πr ALL problems must be completed by the process shown in Example A!! Example A: If a circle has a diameter of 3 meters, what is it’s circumference? Example B: If a circle has a circumference of 12π meters, what is it’s radius? C d C 3 C 3 C 9.4 m C 2 r 12 2 r 12 2 r 2 2 r 6m write the formula substitute simplify (pi written last, like a variable) Only an approximate answer! write the formula substitute divide to get r. simplify Example C: If a circle has a circumference of 20 meters, what is its diameter? C d 20 d 20 d write the formula substitute accurate answer d 6.366 m approximate answer S. Stirling Page 8 of 12 Ch 6 Note Sheet L1 Shortened Key.doc Lesson 6.6 Around the World Read top of page 335. Name ___________________________ You will need to understand units, fractions and multiplication of fractions to get a grip on this section. (P.S. You’ll thank me later when you take Chemistry or Physics too!!) Concept 1: You may divide out common factors when multiplying fractions…before actually multiplying the numerators and then the denominators to get your answer. Warning! You must divide out the same factor from the top and the bottom. (You’re making 1s.) Example A: Multiply. 2 7 15 5 8 21 Example B: Multiply. 1 4 3 7 16 4 5 4 8 14 3 5 Concept 2: You may use the technique of dividing out common factors on units as well. This is called “dimensional analysis”. Here are the key ideas: Example C: How many yards are there in 182.54 centimeters? Idea 1: A fraction like 5 1 5 So since 1 yard 3 feet 1 1 yard = 3 feet, so 3 feet 1 yard 12 inches 1 foot 1 12 inches = 1 foot, so 1 foot 12 inches 1 inch 2.54 cm 1 1 inch = 2.54 cm, so 2.54 cm 1 inch Idea 2: If you multiply a value by 1, you don’t change its value. So if I multiply by fractions that equal 1, I don’t change the quantity just the look of it. Idea 3: You may divide out common factors …even the units! Make sure you set up your fractions so that the units will divide out. 182.54 cm 1 in 1 ft 1 yd 1 2.54 cm 12 in 3 ft Now multiply the numerators together and then the denominators together, then divide the numerator by the denominator. 182.54 yd 1.996 yd 91.44 Example D: You traveled 260 km in 2 hours, what is your mph? How many miles would you go in a minute? (Note: 1 mile = 1.609 km and the word “per” indicates division) 260 km 1 mi 80.8 mph 2 hr 1.609 km S. Stirling 80.8 mi 1 hr mi 1.347 1 hr 60 min min Page 9 of 12 Ch 6 Note Sheet L1 Shortened Key.doc Name ___________________________ EXAMPLE page 341. If the diameter of earth is 8000 miles, find the average speed in miles per hour Phileas Fogg needs to circumnavigate Earth about the equator in 80 days. (Remember distance = rate times time.) Summarize Problem: d = 8000 mi, 80 days, around the equator. Find speed in mph. C d C 8000 miles average speed = mph 24 hr 80 day 1920 hrs 1 day 8000 mi 13.09 mph 1920 hrs OR 80 day 24 hr 1920 hrs 1 day 8000 mi 1 day mi 13.09 80 day 24 hr hr OR for an exact answer… 8000 mi 1 day = 80 day 24 hr 100 mi 1 = 1 24 hr 25 mi 6 hr Speedy Ms. A was running the track at the high school. The track is pictured below. If she makes it around the track 6 times in 6.5 minutes, what was her average speed in meters per second. Keep two decimal places! 112 m C 2 112 2 28 C 399.93 28 m 399.93 6 2399.58 meters 2399.58 m 1 min m 6.15 6.5 min 60 sec sec S. Stirling Page 10 of 12 Ch 6 Note Sheet L1 Shortened Key.doc Lesson 6.7 Arc Length Investigation 10: See Worksheet page 14. Name ___________________________ Measure of an Arc: The measure of an arc equals the measure of its central angle, measured in degrees. Arc length: The portion of (or fraction of) the circumference of the circle described by an arc, measured in units of length. Arc Length Conjecture The length of an arc equals the measure of the arc divided by 360° times the circumference. It is a fraction of the circle! So……. Use the formula every time!! Arc Length = arc degrees Circumference 360 Example (a): Given central ATB 90 with Example (b): Given central COD 180 with radius 12. Find the length of AB . diameter 15 cm. Find the length of CED . B 90 mAB 90 fraction of circle = length of AB 90 1 360 4 90 T 12 mCED 180 A 90 2 12 360 1 24 6 4 180 1 fraction of circle = 360 2 180 length of CED 15 360 15 cm 2 D Example (c): Given mEF 120 with radius = 9 ft. Find the length of EF . 120 F fraction of circle = 120 1 360 3 1 2 9 3 6 ft O 9 E length of EF S. Stirling Page 11 of 12 C O E Ch 6 Note Sheet L1 Shortened Key.doc Name ___________________________ Example (d): If the radius of a circle is 24 cm and mBTA 60 , what is the length of AB ? mAB 120 Example (e): The length of ROT is 116π, what is the radius of the circle? O B 120 60 mROT 360 120 240 R A 120 T degree Circumference 360 120 length of AB 2 24 360 16 50.3 cm arc length S. Stirling arc length degree Circumference 360 240 2 r 360 4 116 r 3 116 3 r then 1 4 r 87 116 Page 12 of 12 T