Survey
* Your assessment is very important for improving the workof artificial intelligence, which forms the content of this project
* Your assessment is very important for improving the workof artificial intelligence, which forms the content of this project
Rational trigonometry wikipedia , lookup
Analytic geometry wikipedia , lookup
Pythagorean theorem wikipedia , lookup
Lie sphere geometry wikipedia , lookup
Trigonometric functions wikipedia , lookup
Problem of Apollonius wikipedia , lookup
Geometrization conjecture wikipedia , lookup
History of trigonometry wikipedia , lookup
Line (geometry) wikipedia , lookup
Tangent lines to circles wikipedia , lookup
History of geometry wikipedia , lookup
Geometry 2205 Unit 4: Mrs. Bondi Geometry Unit 4: Circles Unit 4 Circles Topics: Lesson 1: Lesson 2: Lesson 3: Lesson 4: Lesson 5: Lesson 6: Lesson 7: Lesson 8: Circles and Arcs (PH text 10.6) Areas of Circles, Sectors, and Segments of Circles (PH text 10.7) Geometric Probability (PH text 10.8) Tangent Lines (PH text 12.1) Chords and Arcs (PH text 12.2) Inscribed Angles (PH text 12.3) Angle Measures and Segment Lengths (PH text 12.4) Circles in the Coordinate Plane (PH text 12.5) 1 Geometry 2205 Unit 4: Mrs. Bondi Lesson 1: Circles and Arcs (PH text 10.6) Objectives: to find the measures of central angles and arcs of circles. to find the circumference of a circle and the length of an arc Circle: the set of all points equidistant from a given point called the ______________ Circles are named using the symbol “ ” and the center point. Radius: Diameter: Congruent Circles: Central Angle: The center P of the circle is the midpoint of the diameter! Remember the midpoint formula: Example 1: A diameter of a circle has endpoints A(-3, -2) and B(1, 4). Find the coordinates of the center and find the length of the radius. 2 Geometry 2205 Unit 4: Mrs. Bondi Arc: Arc notation: Three Types of Arcs: Semicircle: Minor Arc: Major Arc: Named by: Named by: Named by: Measure: Measure: Measure: Example: Find each arc in the diagram to the right and its measure. Adjacent Arcs: Postulate 10-2 Arc Addition Postulate: The measure of the arc formed by two adjacent arcs is the sum of the measure of the two arcs. mABC mAB mBC Example 2: L 35̊ P 165̊ 15̊ M N Find the measure of each arc: a. MN b. LM d. LKN c. KLM K 3 Geometry 2205 Unit 4: Mrs. Bondi Circumference - = pi = Theorem 10-9 Circumference of a Circle The circumference of a circle is times the diameter. C d = ratio of circumference to diameter of a circle C d or C 2 r Example 3: The diameter of a circle is 16 cm. Find the circumference. (round to tenth) Example 4: The radius of a circle is 4 in. Find the circumference. (round to tenth) Concentric Circles – Example 5: Compare circumferences d = 12 r = 4 cm c m Arc Length – Theorem 10-10 Arc Length The length of an arc of a circle is the product of the ratio measure of the arc 360 and the circumference of the circle. O length of AB r mAB 2 r 360 B A 4 Geometry 2205 Unit 4: Mrs. Bondi Example 6: If a 60 , diameter = 12, find length AB . (round to nearest tenth) O a B A Example 7: C B If radius = 12 cm, ACB = 210, find length ACB . (round to nearest tenth) A Congruent Arcs - Practice: HW: p.654 #8-34 even 5 Geometry 2205 Unit 4: Mrs. Bondi 6 Geometry 2205 Unit 4: Mrs. Bondi 7 Geometry 2205 Unit 4: Mrs. Bondi Lesson 2: Areas of Circles, Sectors, and Segments of Circles (PH text 10.7) Objective: to find the areas of circles, sectors and segments of circles Theorem 10-11 Area of a Circle The area of a circle is the product of and the square of the radius. A r2 Example 1: The radius of a circle is 15 cm. Find the area. A Sector of a Circle – region bounded by two radii and their intercepted arc. B r Theorem 10-12 Area of a Sector of a Circle The area of a sector of a circle is the product of the ratio measure of the arc 360 and the area of the circle. Area of sector AOB Example 2: The diameter of a circle is 8.2 m, and m AB = 125. Find the area of sector ADB. Round to the nearest tenth. Example 3: Find the area of sector GPH Leave your answer in terms of B C mAB r 2 360 D A 8 . Geometry 2205 Unit 4: Mrs. Bondi Segment of a circle – The part of a circle bounded by an arc and the segment joining its endpoints. Example 4: A circle has radius 8 cm. Find the area of a segment of the circle bounded by a 120 degree arc. Round your answer to the nearest tenth. 120̊ 8 cm Example 5: What is the area of the shaded segment shown at the right? Round your answer to the nearest tenth. 9 Geometry 2205 Unit 4: Mrs. Bondi Practice: HW: p.663 #6-30 even 10 Geometry 2205 Unit 4: Mrs. Bondi 11 Geometry 2205 Unit 4: Mrs. Bondi 12 Geometry 2205 Unit 4: Mrs. Bondi Lesson 3: Geometric Probability (PH text 10.8) Objective: to use segment and area models to find the probability of events Geometric probability uses geometric figures to model occurrences of real-life events. The occurrences can be compared by comparing the measurements of the figures. Reminder: Probability is … P(event) = # of favorable outcomes # of possible outcomes Examples: 1) 2. 3. KYW gives a weather update every 10 minutes. If you turn on the radio at a random time, what is the probability that you would wait more than 4 minutes to hear the weather update? P(wait > 4 min.) = P(wait ≤ 2 min.) = 13 Geometry 2205 Unit 4: Mrs. Bondi 4. Bill takes the train to work in Center City each morning. Because of traffic, he cannot be sure exactly when he will arrive at the station. If trains leave the station every 20 minutes during rush hour, what is the probability he will not need to wait more than 5 minutes for a train to leave? Examples: 5. 6. 7. What is the probability of a tossed coin landing in the shaded region of the rectangle below? The rectangle measures 125 cm long and 25 cm wide. 14 Geometry 2205 Unit 4: Mrs. Bondi 8. Rebekah and Bridget made a dodecagon shaped dartboard for a school carnival. It is divided into a series of triangles by connecting opposite vertices alternately colored red, orange, yellow, green, blue and purple. a) What is the probability that the dart will land in a yellow triangle? b) What is the probability that the dart will land in a red or orange triangle? c) What is the probability that the dart will land in a primary color triangle? d) What is the probability that the dart will land in a triangle whose color name contains the letter r? Practice: HW: p.671 #8-40 even 15 Geometry 2205 Unit 4: Mrs. Bondi 16 Geometry 2205 Unit 4: Mrs. Bondi 17 Geometry 2205 Unit 4: Mrs. Bondi Lesson 4: Tangent Lines (PH text 12.1) Objective: to use properties of a tangent to a circle A B Point of Tangency Tangent to a circle A Tangent to a Circle Point of Tangency – Theorem 12-1 If a line is tangent to a circle, then it is perpendicular to the radius drawn to the point of tangency. If AB is tangent to Example 1: O at P, then AB OP . Example 2: Example 3: What is the distance to the horizon that a person can see on a clear day from an airplane 2 miles above the earth? The Earth’s radius is about 4000 mi. Theorem 12-2 (Converse of 12-1) If a line in the same plane as a circle is perpendicular to a radius at its endpoint on the circle, then the line is tangent to the circle. If AB OP at P, then AB is tangent to 18 O. Geometry 2205 Unit 4: Mrs. Bondi Example 4: Find the length of the radius if PQ = 8 and TQ = 5. P 8 Example 5: Find the length of the segment connecting the centers of the circles. 20 in Q 5 T 6 in 10 in S Theorem 12-3 If two tangent segments to a circle share a common endpoint outside the circle, then the two segments are congruent. If AB and CB are tangent to respectively, then AB CB . O at A and C, The sides of the triangle to the right are tangent to the circle. The circle is “inscribed in” the triangle. The triangle is “circumscribed about” the circle. O is inscribed within ∆PQR. Label each length that you know based on Thm 12-4. Example 5: Q 15 cm X P O Y If ∆PQR has a perimeter of 88cm, find QY. Z 17 cm R 19 Geometry 2205 Unit 4: Mrs. Bondi Practice: HW: p.766 #5-19 20 Geometry 2205 Unit 4: Mrs. Bondi 21 Geometry 2205 Unit 4: Mrs. Bondi 22 Geometry 2205 Unit 4: Mrs. Bondi Lesson 5: Chords and Arcs (PH text 12.2) Objective: to use congruent chord, arcs and central angles to use perpendicular bisectors to chords P Chord – a segment with endpoints on a circle – labeled as a line segment, PQ O – marks an arc on the circle, labeled PQ Q Theorem 12-4 and its converse In the same circle or in congruent circles, a) congruent central angles have congruent arcs b) congruent arcs have congruent central angles Theorem 12-5 and its converse In the same circle or in congruent circles, a) congruent central angles have congruent chords b) congruent chords have congruent central angles Theorem 12-6 and its converse In the same circle or in congruent circles, a) congruent chords have congruent arcs b) congruent arcs have congruent chords B D Let’s prove Theorem 12-6 part a. Given: O P BC DF Statements O Prove: BC DF Reasons 23 P C F Geometry 2205 Unit 4: Mrs. Bondi Theorem 12-7 and its converse In the same circle or in congruent circles, c) chords equidistant from the center(s) are congruent d) congruent chords are equidistant from the center(s) Example 1: answer. Find the value of x in the diagram to the right. Justify your Theorem 12-8 In a circle, if a diameter is perpendicular to a chord, then it bisects the chord and its arc. Theorem 12-9 In a circle, if a diameter bisects a chord (that is not a diameter), then it is perpendicular to the chord. Theorem 12-10 In a circle, the perpendicular bisector of a chord contains the center of the circle. 24 Geometry 2205 Unit 4: Mrs. Bondi Example 2: Let chord CE 30 in. and 6 in. from the center. A 6 in T 30 in C E A a) Find the radius of Example 3: a) A b) Find mCE T C Let chord AB 22 in. and 7 in. from the center. Find the radius of C C A b) E D B Find the length of AB C A Practice: 25 D B Geometry 2205 Unit 4: Mrs. Bondi 3. What is the missing length to the nearest tenth? HW: p.776 #5-15, 30-32 26 Geometry 2205 Unit 4: Mrs. Bondi 27 Geometry 2205 Unit 4: Mrs. Bondi 28 Geometry 2205 Unit 4: Mrs. Bondi Lesson 6: Inscribed Angles (PH text 12.3) Objective: to find the measure of an inscribed angle to find the measure of an angle formed by a tangent and a chord A C inscribed angle AB intercepted arc of C O B ABC inscribed in O C Theorem 12-10 Inscribed Angle Theorem The measure of an inscribed angle is half the measure of its intercepted arc. mB 1 m AC 2 Example 1: a P mPQT 70 mTS 25 T 70 Find a and b 25 Q S b R 29 Geometry 2205 Unit 4: Mrs. Bondi Example 2: A B F 1) Name a pair of congruent inscribed angles 2) Name a right angle 3) Name a pair of supplementary angles C E D Example 3: 107 Find the value of each variable. z x y 98 o 73 Example 4: Find the value of each variable. 30 Geometry 2205 Unit 4: Mrs. Bondi Theorem 12-12 The measure of an angle formed by a chord and a tangent (that intersect on a circle) is half the measure of the intercepted arc. A Example 5: Find the value of each variable. y x C z 58 B Practice: D HW: p.784 #6-18, 20-21, 23-25 31 Geometry 2205 Unit 4: Mrs. Bondi Extra Practice: Good for Review: Mid-Chapter quiz p.788 32 Geometry 2205 Unit 4: Mrs. Bondi 33 Geometry 2205 Unit 4: Mrs. Bondi 34 Geometry 2205 Unit 4: Mrs. Bondi Lesson 7: Angle Measures and Segment Lengths (PH text 12.4) Objective: to find the measures of angles formed by chords, secants and tangents. to find the lengths of segments associated with circles Secant – a line (or segment, or ray) that intersects a circle at two points – a secant always contains a chord Theorem 12-13 The measure of an angle formed by two lines (or chords) that intersect inside a circle is half the sum of the measures of the intercepted arcs. m1 1 x y 2 Theorem 12-14 The measure of an angle formed by two lines that intersect outside a circle is half the difference of the measures of the intercepted arcs. m1 1 x y 2 Examples: Find the value of the variables. 55 1. 2. 35 103 3. 4. 130 90 170 x y 35 39 y 35 x Geometry 2205 Unit 4: Mrs. Bondi B 5. 6. z 72 A x 55 80 C D Theorem 12-15 Summary: Case I: The products of the chord segments are equal Case II: The products of the secants and their outer segments are equal Case III: The product of a secant and its outer segment equals the square of the tangent Examples: 7. 8. 16 x x 7 9 12 11 5 9. 10. What is the length of AB ? 15 10 12 z A B 3 36 Geometry 2205 Unit 4: Mrs. Bondi 11. Find the value of the variables. 2 y 3 11 6 x 8 Practice: HW: p.794 #8-20, 24-26 37 Geometry 2205 Unit 4: Mrs. Bondi Extra Practice: 38 Geometry 2205 Unit 4: Mrs. Bondi 39 Geometry 2205 Unit 4: Mrs. Bondi 40 Geometry 2205 Unit 4: Mrs. Bondi 41 Geometry 2205 Unit 4: Mrs. Bondi Lesson 8: Circles in the Coordinate Plane (PH text 12.5) Objective: to write the equation of a circle to find the center and radius of a circle Think about some points that would make this equation true. x 2 y 2 25 Imagine a compass connecting these points in a circular shape. Theorem 12-16 The standard form of an equation of a circle with center (h, k) and radius r is x h 2 y k 2 r 2 This is easily derived from the distance formula. d x2 x1 Examples: 2 y 2 y1 2 Find the center and radius. 1. x2 + y2 = 9 2. x2 + y2 = 36 3. x2 + y2 = 121 4. (x – 6)2 + (y – 2)2 = 25 5. (x – 8)2 + (y + 4)2 = 225 6. (x + 3)2 + (y + 7)2 = 100 Example 7: Write the standard equation of each circle. a) center (3, -4) and radius 6 b) 42 center (-2, -1) and radius 2 Geometry 2205 Unit 4: Mrs. Bondi Using Algebra: When given the center and a point on the circle, … The distance formula can be used to find the circle’s radius. The midpoint of the diameter is the center of the circle. What is the standard equation of the circle with center (1, -3) that passes through the point (2, 2). distance formula to find radius: Use radius (computed) and center (given) to write an equation. Example 8: A diameter of a circle has endpoints (-3, 7) and (5, 5). Write an equation of the circle. Practice: HW: p.801 #8-38 even 43 Geometry 2205 Unit 4: Mrs. Bondi 44 Geometry 2205 Unit 4: Mrs. Bondi 45 Geometry 2205 Unit 4: Mrs. Bondi Circle Relationships Visualize at a glance the relationships between the various angles, arcs, and sectors of a circle. 46