GEO: Unit 8 Circles 12.1 Lines that Intersect Circles (1)

Transcription

1 GEO: Unit 8 Circles 12.1 Lines that Intersect Circles (1) NAME Objectives: Identify tangents, secants, and chords. Use properties of tangents to solve problems. This photograph was taken 216 miles above Earth. From this altitude, it is easy to see the curvature of the horizon. Facts about circles can help us understand details about Earth. Recall that a circle is the set of all points in a plane that are from a given point, called the of the circle. A circle with center C is called circle C, or C. The of a circle is the set of all points inside the circle. The of a circle is the set of all points outside the circle. Example 1: Identify each line or segment that intersects L. chords: secant: diameter: tangent: point of tangency: radii:

2 Example 2: Identify each line or segment that intersects P. chords: secant: diameter: tangent: point of tangency: radii: Example 3: Find the length of each radius. Identify the point of tangency and write the equation of the tangent line at this point. Radius of S = Radius of R = Point of tangency = Equation of tangent line:

3 A is a line that is tangent to two circles. Example 4: Early in its flight, the Apollo 11 spacecraft orbited Earth at an altitude of 120 miles. What was the distance from the spacecraft to Earth s horizon rounded to the nearest mile? Draw and label a diagram to support your answer. (It is 4000 miles from the center of the earth to the surface.radius.)

4 Example 5: Kilimanjaro, the tallest mountain in Africa, is 19,340 ft tall. What is the distance from the summit of Kilimanjaro to the horizon to the nearest mile? Draw and label a diagram to support your answer. (Miles and feet convert to miles ft = 1 mile) Example 6: RS and RT are tangent to Q. Find RS HOMEWORK: page 797 #1-4, 6-11, 13-27, 31.

5 12.2 Notes Arcs and Chords (2) Objectives: Apply properties of arcs. Apply properties of chords. A is an angle whose vertex is the center of a circle. An is an unbroken part of a circle consisting of two points called the endpoints and all the points on the circle between them. Minor arcs may be named by two points. Major arcs and semicircles must be named by three points.

6 Example 1: The circle graph shows the types of grass planted in the yards of one neighborhood. Find mklf. Example 2: Use the graph to find each of the following. a. m FMC b. mahb c. m EMD are arcs of the same circle that intersect at exactly one point. RS and ST are adjacent arcs.

7 Example 3: Find mbd Within a circle or congruent circles, are two arcs that have the same measure. In the figure ST UV.

8 Example 4: TV WS. Find mws. Example 5: C J, and m GCD m NJM. Find NM.

9 Example 6: Find NP. (RS = 8 and SM = 9.) Example 7: Find ZY. Round to the nearest hundredth HOMEWORK: page 806 #1-28, 31, 38, 39, 51

10 Objectives: Find the area of sectors. Find arc length Sector Area and Arc Length (3) Warm up: 1. What is the area of a rectangle 12 in long and 6 in wide? 2. What is the area of half of the rectangle in number 1? 3. What is the area of ¾ of the rectangle in number 1? 4. What is the area of a circle with radius of 8 miles? 5. What is the area of half of a circle with radius of 6 kilometers?

11 Example 1: Find the area of each sector. Give answers in terms of π and rounded to the nearest hundredth. sector HGJ Example 2: Find the area of each sector. Give answers in terms of π and rounded to the nearest hundredth. sector ABC Example 3: Find the area of each sector. Give your answer in terms of π and rounded to the nearest hundredth. sector ACB Example 4: A windshield wiper blade is 18 inches long. To the nearest square inch, what is the area covered by the blade as it rotates through an angle of 122? In the same way that the area of a sector is a fraction of the area of the circle, the length of an arc is a fraction of the circumference of the circle.

12 Example 5: Find each arc length. Give answers in terms of π and rounded to the nearest hundredth. Find FG. Example 6: Find each arc length. Give answers in terms of π and rounded to the nearest hundredth. An arc with measure 62 o in a circle with radius 2 m Example 7: Find each arc length. Give your answer in terms of π and rounded to the nearest hundredth. Find GH HOMEWORK: page 813 #1-5, 9-11, 13, 15, 23, 26, 30, 32.

13 12.4 Inscribed Angles (5) Objectives: Find the measure of an inscribed angle. Use inscribed angles and their properties to solve problems. An is an angle whose vertex is on a circle and whose sides contain chords of the circle. An consists of endpoints that lie on the sides of an inscribed angle and all the points of the circle between them. A chord or arc an angle if its endpoints lie on the sides of the angle. Example 1: Find each measure. m PRU msp Example 2: Find each measure. m DAE madc

14 Example 3: Find a. Example 4: Find m LJM. Example 5: Find z.

15 Example 6: Find m EDF. Example 7: Find the angle measures of quadrilateral GHJK. Example 8: Find the angle measures of quadrilateral JKLM HOMEWORK: page 824 #1-11, 16, 17-19, 21, 26.

16 Angle Relationships in Circles (6) Objectives: Find the measures of angles formed by lines that intersect circles. Use angle measures to solve problems. Example 1: Using Tangent-Secant and Tangent-Chord Angles Find each measure. m EFH mgf I Example 2: Using Tangent-Secant and Tangent-Chord Angles Find each measure. m STU msr

17 Finding Angle Measures Inside a Circle Example 3: Find each measure. m AEB m AED Example 4: Find each angle measure. m ABD m ABE Example 5: Find each angle measure. m RNM

18 Finding Measures Using Tangents and Secants Example 6: Find the value of x. Example 7: Find the value of x. Example 8: In the company logo shown, mfh 108, and mlj 12 What is m FKH? Example 9: Two of the six muscles that control eye movement are attached to the eyeball and intersect behind the eye. If maeb = 225, what is m ACB?

19 Angle Relationships in Circles Where is the vertex of the angle? What is the measure of the angle? Draw a diagram to depict the angle and circle. Conclusion: Finding Arc Measures Example 10: Find myz Example 11: Find mlp 12.5 HOMEWORK: page 834 #1-17, 21, 25,

20 12.7: Circles in the Coordinate Plane (7) Objectives: Write equations and graph circles in the coordinate plane. Use the equation and graph of a circle to solve problems. REVIEW: What is the distance formula? Use that formula to find the distance between (2, 2) and (5, 7). Example 1: Write the equation of each circle below: a) J with center J(2, 2) and radius 5. b) P with center P(0, 3) and radius 8 c) K that passes through J(6, 4) and has center K(1, 8) d) Q that passes through (2, 3) and has center Q(2, 1)

21 What is the connection between the distance formula above and the equation of the circle? y x If you are given the equation of a circle, you can graph the circle by making a table or by identifying its center and radius. Graph x 2 + y 2 = 16. Step 1 Step 2 y OR YOU CAN x Example 2: For each of the following circle equations, list the center and the radius. 1. ( x h) ( y k) r C = r = x y C = r =

22 ( x 3) ( y 3) 4 4. C = r = 2 2 ( x 3) ( y 3) 1 C = r = 5. x ( y 2) ( x 7) y 9 C = C = r = r = Graph (x 3) 2 + (y + 4) 2 = 9. Graph x² + y² = 9. y y x x Graph (x 3) 2 + (y + 2) 2 = 4 y x 12.7 HOMEWORK: page 850 #1-8, 10-17, 19, 20, 22-26, 30,