Arcs and Angles of Circles

Unit 8 Arcs and Angles of Circles Lesson 1: Thales’ Theorem Draw a diagram for each of the vocabulary words. Definition Circle • The set of all points equidistant from a given point Radius • A segment that joins the center of the circle with any point on the circle Diameter • A segment that passes through the center and whose endpoints are on the circle Chord • A segment whose endpoints are on the circle Central Angle • An angle whose vertex is on the center of the circle Semicircle • Half a circle formed by a diameter Diagram Thales’ Theorem Discovery Activity You will need a colored index card. a. Take the colored index card provided and push the card between points A and B pictured below: b. Mark on your paper the location of the corner of the colored index card and label this as point C. (make sure the sides of the index card are always touching A and B) c. Do this again, pushing the corner of the colored index card up between A and B but at a different angle. Again, mark the location of the corner, labeling it as point D. d. Continue locating points in the same manner in all directions through A and B, labeling the points as you go (create at least 8 eight points). • What shape do the points create? • Connect points A and B. What have you created? • Draw in ∠ACB . What type of angle is ∠ACB ? • What type of triangle is ΔACB ? Thales’ theorem is the relationship between the diameter and the points on a circle listed formally below. Thales’ Theorem: If A, B, and C are three distinct points on a circle and segment the circle, then is a right angle. is a diameter of Example 1 You will need a compass and a straightedge • • • • • Draw a circle with center P. Draw diameter AB . Label point C anywhere on the circumference of the circle. Draw ΔAPC . Draw ΔBPC . a. What type of triangles are ΔAPC and ΔBPC ? How do you know? b. Explain why ∠ACB is a right angle. Example 2 You will need a compass and a straightedge • • • Draw a circle with center P. Draw diameters AC and BD of the circle. Connect the endpoints of the diameters to form a rectangle. a. Explain why this shape will always be the result. b. What are the two properties of the diagonals of a rectangle? Exercise 1 AB is the diameter of the circle shown. The radius is 12.5 cm , and AC = 7 cm . a. Find m∠C b. Find AB c. Find BC Exercise 2 In the circle shown, BC is the diameter with center A. a. Find m∠DBA b. Find m∠BEA c. Find m∠DAB d. Find m∠BAE e. Find m∠DAE Homework 1. Determine the length of the radius of the circumscribed circle to the right triangle with legs 7 cm and 4 cm. Round your answer to the nearest hundredth. 2. In the figure below, AB is the diameter of a circle of radius 17 miles. If BC = 30 miles, what is AC ? 3. Explain why there is something mathematically wrong with the picture below. 4. In the figure below, O is the center of the circle, AD is a diameter and m∠DBO = 24° . a. Find m∠BDO . b. Find m∠BOD . c. Find m∠AOB . d. If m∠AOB : m∠BOC = 3: 4 , what is the m∠BOC ? 6 Lesson 2: Circles, Chords, Diameters, and Their Relationships You will need a compass and a straightedge Opening Exercise a. Construct a circle of any radius and identify the center as point P. b. Draw a chord, and label it AB c. Construct the perpendicular bisector of AB d. What do you notice about the perpendicular bisector of AB ? Example 1 Using the construction above: a. Draw another chord and label it CD b Construct the perpendicular bisector of CD c. What do you notice about the perpendicular bisector of CD ? d. What do we know about any point along the diameter(s) in relation to the endpoints of the chord? e. Based on our answer in part d, what is special about point P? Example 2 Prove the theorem: Congruent chords define central angles equal in measure. Given: Prove: Example 3 Prove the theorem: If a diameter of a circle bisects a chord, then it must be perpendicular to the chord. Given: Circle C with diameter DE , chord AB and AF = BF . Prove: DE ⊥ AB Exercises 1. In circle B, BE ⊥ AC , AB = 10 , and AC = 16 . Find DE . 2. In circle G, AC = 24 and DG = 13 . Find EG . Explain your work. (Hint: Draw in AG .) 3. In the figure, the two circles have equal radii and intersect at points B and D. A and C are centers of the circles. If BD ⊥ AC , AC = 8 , and the radius of each circle is 5, find BD . (Hint: Draw in BA and BC .) Homework 1. Given circle A shown, AF = AG and BC = 22 . Find DE . 2. In the figure, circle P has a radius of 10 and AB ⊥ DE . If AB = 8 , what is the length of AC ? 3. In the figure, circle P has a radius of 10 and AB ⊥ DE . If DC = 2 , what is the length of AB ? 4. In circle O, AB = 30 , OM = 20 , and ON = 18 . What is CN to the nearest hundredth? 5. Given: Circle O with chords AB and CD ∠AOB ≅ ∠DOC Prove: AB ≅ CD Lesson 3: Rectangles Inscribed in Circles Opening Exercise a. Given circle D with a radius of 17, AB = 30 and AB ⊥ DE . Find DE. b. In circle F, CF ⊥ AE , CF = 8 and the two concentric circles have radii of 10 and 17. Find DE . Example 1 You will need a compass and a straightedge • Draw circle P. • Draw a right triangle inscribed in the circle with the diameter being the hypotenuse of the right triangle. • Construct the image of the right triangle after a rotation of 180° about the center of the circle. • What kind of figure is formed? Example 2 You will need a compass and a straightedge • Draw circle P. • Draw a right triangle inscribed in the circle with the diameter being the hypotenuse of the right triangle. • Construct the image of the triangle after the reflection over the diameter. • What kind of figure is formed? Example 3 You will need a compass and a straightedge Finding the center of a given circle! • Draw chord AB • Construct the perpendicular bisector to AB • Draw chord CD • Construct the perpendicular bisector to CD • Identify the point of intersection of the two perpendicular bisectors. You found the center of the circle!!! Example 4 You will need a compass and a straightedge Construct a square inscribed in a circle. Exercises 1. 2. ΔABD was reflected across diameter BD to create kite ABED. Find the measure of the following angles if m∠ADB = 40° . a. m∠BDE b. m∠BAD c. m∠BED d. m∠ABD e. m∠EBD f. m∠ABE g. m∠ADE In circle A, DF and BG are parallel chords 14 cm apart. If DF = 12 cm , AB = 10 cm , and EH ⊥ BG , find BG . Homework 1. Rectangle ABCD is inscribed in circle P . Boris says that the diagonal AC could pass through the center. Is Boris correct? Explain your answer in words or draw a picture to explain your reasoning. 2. In the figure, BCDE is a rectangle inscribed in circle A . If DE = 8 and BE = 12 , find AE in simplest radical form. 3. Given circle A, BC = CD = 8 and AD = 13 . Find the radius of the circle in simplest radical form. Lesson 4: Experiments with Inscribed Angles Opening Exercise Draw a diagram for each of the vocabulary words. Definition Arc • A portion of the circumference of a circle Inscribed Angle • An angle whose vertex is on the circle, and each side of the angle intersects the circle in another point Central Angle • An angle whose vertex is the center of the circle Minor Arc • An arc of a circle having a measure less than 180 degrees Major Arc • An arc of a circle having a measure greater than 180 degrees Intercepted Arc • The arc cut in the circle by an inscribed or central angle Semicircle • Half a circle formed by a diameter Diagram Example 1 Given circle A, identify the following using the vocabulary from the Opening Exercise: a. ! BE b. ! CDE c. ! EDF d. ! FED e. ∠BAE f. ∠BDC g. ∠ECF Example 2 You will need a straightedge a. Draw a point on the circle, and label it D. b. Create ∠BDC . c. ∠BDC is called an inscribed angle. Explain why. d. ! is called the intercepted arc. BC Explain why. Example 3 You will need a straightedge and protractor a. Draw a point on the circle in a different location than you did in Example 2, and label it E . b. Create ∠BEC . c. Compare your angles from Example 2 and Example 3. d. What appears to be true about ∠BDC and ∠BEC ? e. Confirm your theory about ∠BDC and ∠BEC by measuring them with the protractor. f. What conclusion may be drawn from this? Example 4 You will need a straightedge and protractor a. Draw the angle formed by connecting points B and C to point A, the center of the circle. b. Is ∠BAC an inscribed angle? Explain. c. Is ∠BAC a central angle? Explain. d. What is the intercepted arc? e. Measure ∠BAC with a protractor. Is m∠BAC the same as one of the inscribed angles in Examples 2 and 3? f. Make a prediction about the relationship between the inscribed angle and the central angle. Exercise 1 Using a protractor, measure both the inscribed angle and the central angle shown on circle A below. m∠BCD = ___________ m∠BAD = ___________ Exercise 2 Using a protractor, measure both the inscribed angle and the central angle shown on the central angle shown on circle A below. m∠BAC = ___________ m∠BDC = ___________ Summary When inscribed angles and central angles share the same intercepted arc: • • The inscribed angle is _______________________ the measure of the central angle. The central angle is _______________________ the measure of the inscribed angle. Exercise 3 In circle O, AC is the diameter, m∠COD = 120° , and BD bisects ∠ADO . Find the following and explain. a. m∠AOD b. m∠OAD c. m∠BDA d. m∠BEC e. m∠ACD f. m∠ABD 1. Using circle A pictured to the right, give an example of the following: Minor Arc: Major Arc: Inscribed Angle: Central Angle: 2. What is the relationship between the measure of the inscribed angle and the measure of the central angle that intercepts the same arc? 3. Solve for the value of x in each of the following circle with centers at D: a. b. Lesson 5: Inscribed Angle Theorem Opening Exercise In each of the following diagrams of circle O, the measure of ∠COA = 50° , find m∠CBA . Inscribed Angle Theorem The measure of a central angle is ____________________________ the measure of any inscribed angle that intercepts the same arc as the central angle. Consequence of Inscribed Angle Theorem Inscribed angles that intercept the same arc are ____________________________. Example 1 Find the value of x in each of the diagrams below. Point A represents the center of each circle. a. m∠D = 25° b. m∠B = 32° x x c. m∠D = 15° d. x m∠D = 19° x 26 Exercises Find the value of x (and y in part d) in each of the diagrams below, assuming the point at the center is the center of the circle. 1. 2. x 3. x 4. x Example 2 Find the value of x in circle A. Explain. Example 3 Find the value of x in circle A. Explain how you found your answer. Example 4 Find the value of x in circle A. Find the value of x in each of the following: 1. AB is the diameter 2. x x 60° is the central angle 4. x A A 3. Lesson 6: Unknown Angle Problems with Inscribed Angles Opening Exercise 1. Find the value of x if AC is a diameter. Explain how you calculated your answer. 2. Is YZ a diameter? Explain your reasoning. Exercises 1. Find the value of x. 2. In circle A, find the value of x if m∠BAD = 62° . 3. Find the measures of angles x and y. Explain the relationships and theorems used. Lesson 7: The Angle Measure of an Arc Opening Exercise In circle A, if the measure of ∠GBF is 17° , name 3 other angles that have the same measure and explain why. What is the measure of ∠GAF ? Explain. Can you find the measure of ∠BAD ? Explain. Example 1 The above circles are similar, with central angles of 70° . Explain how we can prove this. Are all circles similar? Explain. Example 2 The concentric circles all have a center at point O. a. Name the central angle. b. Name three minor arcs. c. Name a major arc. d. Using a protractor, find the measure of ∠AOB . e. !. Find mEF f. !. Find mCD g. !. Find mAB h. Explain how central angles relate to their intercepted arcs. i. What similarity transformation maps all circles to one another? The angle measure of a ______________________________ is the measure of the corresponding ___________________________. Example 3 ! and CD ! adjacent in circle A? Are BC Write a definition for adjacent arcs. !? ! = 25° and CD ! = 35° , what is the angle measure of BD If BC Example 4 ! : CE ! : ED ! : DB ! = 1: 2 : 3 : 4 . Find: In circle A, BC a. m∠BAC b. m∠DAE c. ! mDB d. ! mCED Example 5 In circle B, AB = CD . Find: a. ! mCD b. ! mCAD c. ! mAD 1. 2. Given circle A: Identify: Find the measure of: a. central angle f. ! mBE b. an inscribed angle g. ! mCD c. a chord h. ! mCE d. a minor arc i. ! mBD e. a major arc ! = 2mBD ! find: In circle A, BC is a diameter and m∠DAC = 100° . If mEC a. m∠BAE b. ! mEC c. ! mDEC Lesson 8: Arcs and Chords Opening Exercise Given circle A with BC ⊥ DE , FA = 6 , and AC = 10 . Find BF and DE . Explain your work. (hint: connect AD and AE ) Example1 You will need a straightedge. Prove: If two chords are congruent, the arcs they subtend are congruent. Example 2 ! ! Given circle A with mBC = 54° and ∠CDB ≅ ∠DBE , find mDE . What must be true about BE and CD ? Explain. Theorems Congruent Chords • Congruent chords have congruent arcs. • Congruent arcs have congruent chords. Parallel Chords • Arcs between parallel chords are congruent. Example 3 Prove: Arcs between parallel chords are congruent. Exercises 1. ! and ED !. In circle A find the measures of CD 2. ! : mDE ! : mEC ! = 1: 3 : 5 . Find: BC is a diameter of circle A. mBD 3. a. ! mBD b. ! mDEC c. ! mECB ! = mED ! and mEC ! : mCB ! : mBD ! = 5 : 2 : 3 . Find: mCB a. m∠BCF b. m∠EDF c. m∠CFE 1. 2. 3. If m∠CDE = 35° in circle F, find: a. ! mCE b. ! mBD c. ! mED ! = mED ! , and m∠CAE = 32° . In circle A, BC is a diameter, mCE a. Find m∠CAD . b. Find m∠ADC . ! : mED ! , and In circle A, BC is a diameter, 2mCE BC P DE. Find m∠CDE . Lesson 9: Arc Length and Areas of Sectors Opening Exercise Answer the following for circle C: a. How many degrees make up a full rotation of a circle? b. !? How many degrees are in the measure of AB c. What is the measure of ∠ACB ? d. What kind of angle is ∠ACB ? e. Find the exact value of the circumference of the circle. f. !? What is the exact measure of the length of AB Definition Diagram Arc Length • the circular distance around the arc Example 1 Using circle A, find the exact length of the arc of degree measure 60° in a circle of radius 10 cm. A Formula for Arc Length Example 2 The radius of the pictured circle is 36 cm, and m∠ABC = 60° . !? What is the exact arc length of AC Example 3 a. ! to the In circle A, find the length of arc BC nearest tenth. b. Using the same concept we used to find arc length, how can we find the area of the shaded region to the nearest tenth? Formula for Area of a Sector Example 4 ! with an arc measure of 60° . Sector AOB has an area of 24π . Circle O has a minor arc AB What is the length of the radius of circle O? Exercises 1. The area of sector AOB in circle O is 28π and the radius is 12 cm. Find the measure of ∠AOB . 2. In the following figure, circle O has a radius of 8 cm, m∠AOC = 108° , and 𝐴𝐵 = 𝐴𝐶 = 10 cm. Find: a. m∠OAB b. ! mBC c. Area of sector BOC. 1. 2. a. In circle O find the exact value of the arc !. length of PQR b. Find the exact area of sector POR. Using the picture of circle O shown, determine the following to the nearest tenth: a. ! arc length of PQ b. area of sector POQ Lesson 10: Unknown Length and Area Problems Opening Exercise 1. In circle O, find the exact area of the shaded region. 2. In circle O, find the area of the shaded region to the nearest tenth. Example 1 Another way to measure angles: 1. 2. 3. 4. Draw circle A of any size. Draw in radius AB. Measure the radius using the string provided to you. Using your string as a measuring tool, measure and mark the number of strings needed to go around the circle once. Approximately how many strings did it take to make it all the way around the circle? Was this the same for everyone in the class? What is the relationship between the circumference and radius? What does this really mean? The central angle that intercepts two consecutive string markings on your arc is equal to 1 radian. Definition Diagram Radian • the measure of a central angle when the arc it subtends is equal in length to the radius In Example 1, we saw that it takes 2π radii to go all the way around any circle. ( C = 2π r ) Therefore, 2π radians = 360° . How can we determine the number of degrees there are in 1 radian? Formulas (on Reference Sheet!) Radians Degrees 1 radian = 180 degrees π 1 degree = π radians 180 Example 2 Circle B has a radius of 10 cm. and the measure of central angle B is 1.5 radians. Find the length of the intercepted arc. Example 3 Circle B has a radius of 14 cm. Angle B intercepts the arc with a length of 6π . Find the measure of angle B in radians. 1. 2. Given circle A, find to the nearest hundredth: a. ! in degrees. mBC b. the area of sector BAC. In circle A, find the area of the entire circle given the area of the sector. Lesson 11: Unknown Length and Area Problems Exercises 1. In circle A, find the area of the shaded region to the nearest hundredth if the m∠BAC = 62° . 2. Using circle A and circle B, find the area of the shaded region to the nearest hundredth. A B 3. When designing a postal flyer, 8 congruent rectangular envelopes are arranged in a circular pattern as shown. The radius of the circle is 14 inches and the dimensions of each envelope are 8” by 3”. a. Divide the circle into congruent sectors with each sector containing one envelope, as shown. What is the area, to the nearest hundredth, of the remaining space in each sector after the envelope has been put in place? b. Using your answer from part a, determine the amount of area, to the nearest hundredth, in the circle minus the envelopes. 4. As an advertisement for a new brand of aquatic paint colors, an old ship’s wheel is being attached to a billboard and the openings between the spokes are being filed with a colored plastic representing samples of the aquatic colors (as shown). a. If the radius of the wheel (not considering the handles) is 36 inches and the radius of the inner hub is 12 inches, find the number of square inches, to the nearest hundredth, of colored plastic needed for each sample color. b. A piece of rope of a coordinating color is to be glued to the outer rim of the wheel adjacent to each aquatic color. Find the number of inches, to the nearest hundredth, of rope needed for each color.