What is Unit 3 about?

1. What is Unit 3 about? We will learn to use arcs, angles, and segments in circles to solve real life problems. We will learn how to find the measure of angles related to a circle. We will find circumference and area of figures with circles. We will also find the surface and volume of spheres.

2. Crop Circles Whether you think crop circles are made by little green men from space or by sneaky earthling geeks, you've got to admit that they are pretty dang cool...  And whoever is making them knows a ton of geometry! http://www.coolmath.com/lesson-geometry-of-crop-circles-1.html

3. Properties of Tangents Friday, November 28, 2014 Essential Question: How do we identify segments and lines related to circles and how do we use properties of a tangent to a circle? Lesson 6.1 M2 Unit 3: Day 1

4. 7 ANSWER radius or diameter 13 in. ANSWER ANSWER 4 cm 3 ANSWER ANSWER Warm Ups 1. What measure is needed to find the circumference or area of a circle? 2.Find the radius of a circle with diameter 8 centimeters. 3.A right triangle has legs with lengths 5 inches and12 inches. Find the length of the hypotenuse. 5.Solve (x + 18)2 = x2 + 242. 4.Solve 6x + 15 = 33.

6. Circle Diameter Chord Radius Circle The set of all points in a plane that are equidistant from a given point, called the center. E F Tangent P D A Secant C B Name of the circle: ʘ P

7. Definition Diameter – a chord that contains the center of the circle. • Radius – a segment from the center of the circle to any point on the circle. C A B

8. Definition Chord – a segment whose endpoints are points on the circle.

9. Definition Secant – a line that intersects a circle in two points.

10. Definition Tangent – a line in the plane of a circle that intersects the circle in exactly one point. P Point of tangency O

11. A little extra information The word tangent comes from the Latin word meaning to touch The word secant comes from the Latin word meaning to cut.

12. 1. Tell whether the line or segment is best described as a chord, a secant, a tangent, a diameter, or a radius. EXAMPLE 1 tangent diameter chord radius

13. is a diameter because it is a chord that contains the center C. b. AB 2. Tell whether the line, ray, or segment is best described as a radius, chord, diameter, secant, or tangent of ʘ C. is a tangent ray because it is contained in a line that intersects the circle at only one point. AB AC AC a. c. b. d. c. DE DE AE d. is a secant because it is a line that intersects the circle in two points. AE a. is a radius because Cis the center and Ais a point on the circle. EXAMPLE 2 Identify special segments and lines SOLUTION

14. 3. Use the diagram to find the given lengths. a. a. The radius of ʘ Ais 3 units. Radius of ʘ A b. b. The diameter of ʘ Ais 6 units. Diameter of ʘ A c. c. The radius of ʘ B is 2 units. Radius of ʘ B Diameter of ʘ B The diameter of ʘ Bis 4 units. d. d. EXAMPLE 3 Find lengths in circles in a coordinate plane SOLUTION

15. 4. Find the radius and diameter of ʘ Cand ʘ D. a. The radius of ʘ Cis 3 units. b. The diameter of ʘ Cis 6 units. c. The radius of ʘ D is 2 units. The diameter of ʘ Dis 4 units. d. GUIDED PRACTICE EXAMPLE 4 SOLUTION

16. Definitions Common tangent – a line or segment that is tangent to two coplanar circles Common internal tangent – intersects the segment that joins the centers of the two circles Common external tangent – does not intersect the segment that joins the centers of the two circles

17. 5. Tell whether the common tangents are internal or external. EXAMPLE 5 a. b. common internal tangents common external tangents

18. c. 2 common tangents c. a. b. b. a. 4 common tangents 3 common tangents Draw common tangents EXAMPLE 6 6. Tell how many common tangents the circles have and draw them. SOLUTION

19. Perpendicular Tangent Theorem 6.1 In a plane, if a line is tangent to a circle, then it is perpendicular to the radius drawn to the point of tangency.

20. Tangent Theorems Create right triangles for problem solving.

21. 7. In the diagram, Bis a point of tangency. Find the radiusr of ʘC. SOLUTION You know that AB  BC, so △ABCis a right triangle. You can use the Pythagorean Theorem. EXAMPLE 8 Find the radius of a circle AC2 = BC2 + AB2 Pythagorean Theorem Substitute. (r + 50)2 = r2 + 802 Write the binomial twice. (r + 50)(r + 50) = r2 + 802 Multiply. r2 + 50r +50r + 2500 = r2 + 6400 Combine Like Terms. r2 + 100r + 2500 = r2 + 6400 100r = 3900 Subtract from each side. r = 39 ft. Divide each side by 100.

22. 8. ST is tangent to ʘ Q. Find the value of r. SOLUTION You know from Theorem 10.1 that STQS, so △QSTis a right triangle. You can use the Pythagorean Theorem. EXAMPLE 8 QT2 = QS2 + ST2 Pythagorean Theorem (r + 18)2 = r2 + 242 Substitute. r2 + 36r + 324 = r2 + 576 Multiply. 36r = 252 Subtract from each side. r = 7 Divide each side by 36.

23. Perpendicular Tangent Converse In a plane, if a line is perpendicular to a radius of a circle at its endpoint on the circle, then the line is tangent to the circle.

24. 9. IsDEtangent to ʘ C? ANSWER Yes – The length of CE is 5 because the radius is 3 and the outside portion is 2. That makes ∆CDE a 3-4-5 Right Triangle. So DE and CD are  GUIDED PRACTICE EXAMPLE 10

25. 10. In the diagram, PTis a radius of ʘ P. Is STtangent to ʘ P ? SOLUTION Use the Converse of the Pythagorean Theorem. Because 122 + 352 = 372,△PSTis a right triangle and STPT. So, STis perpendicular to a radius of ʘ Pat its endpoint on ʘ P. STis tangent to ʘ P. Verify a tangent to a circle EXAMPLE 7

26. Congruent Tangent Segments Theorem 6.2 If two segments from the same exterior point are tangent to a circle, then they are congruent.

27. RSis tangent to ʘCat Sand RTis tangent to ʘCat T. Find the value of x. Tangent segments from the same point are 11. SOLUTION RS= RT 28 = 3x + 4 Substitute. 8 = x Solve for x.

28. 12.

29. Homework Page 187 # 18 – 24 all Page 188 # 1 – 10.