Chapter 12 Final Exam Review Project

1. Circles Diameter Radii Pi Arc Sector Degree Measure Chapter 12Final Exam Review Project By: JilliAN Rizzitano AND LAURA SACCO Geometry H Mrs. Liedell

2. Overview:formatted by: Jillian Circles Diameter Radii Pi Arc Sector Degree Measure • Lesson 1(Jillian)—Circles, Radii & Chords • Lesson 2(Jillian)—Tangents • Lesson 3(Jillian)—Central Angles and Arcs • Lesson 4(Jillian)—Inscribed Angles • Lesson 5(Laura)—Secant Angles • Lesson 6(Laura)—Tangent Segments and Intersecting Chords • Extra Lesson 1(Laura)—Arcs and Angles • Extra Lesson 2(Laura)—Segments Π

3. Lessons 1-4 Jillian Π

4. definitions of Circles Review Circle—set of all pts. in a plane that are a given dist. from a given pt. in the plane • Diameter(d)= segment that pass through the center of the circle & whose endpoints are on the circle. • Radius(r)—segment with 1 point on the circle and the other as the center • Circumference of a circle(c)—distance around it • Pi– πis ≈3.1415926535… and is the ratio of the circumference to the diameter of a circle

5. People if you forget this mrs. Liedell might tattoo it to your head (seriously): All radii are =.

6. Practice Problems! What do all these segments have in common? The three segments are equal. center

7. Practice Problems! Quick T/F (copied from Ch. 12 Test): F A radius of a circle is also a chord of a circle. Infinitely many secant lines can be drawn from a specific point outside the circle through the circle. Infinitely many tangent lines can be draw from a specific point outside a circle to the circle. If two chords in a circle intersect at the center, they are each diameters of the circle. Given points A, B, and C on a circle with diameter AC, = 180˚ T F T T

8. What are concentric circles? • Concentric circles—circles are concentric iff they lie in the same plane & have the same center (see below)

9. Thm.s 56-58 Theorems: • If a line passes through the center of a circle & is ⊥ to chord→ bisects chord • If a line passes through center of a circle bisects a chord (that’s not the diameter) → it is also ⊥ to chord • The ⊥ bisector of a chord of a circle → has center of circle

10. Tangent time: Tangent—a line in the plane of the circle that intersects the circle in exactly one point Here’s an example of a tangent: ______________________________________________________________________________Two Tangent Theorems: • If a line is tangent to the circle → the tangent line is ⊥ to radius to the point of contact (the point of tangency) • If a line is ⊥ to a radius at its outer endpoint → it’s tangent to the circle (ALSO: If a line ⊥ to radius at endpoint on O→ it’s a tangent line)

11. Definitions: Just to note: Usually minor arcs are named by 2 points and major arcs & semicircles are named by 3 points.

12. Angles and Arcs • Central ∠—angle whose vertex is the center of a circle • degree measure of an arc(NOT TO BE CONFUSED WITH ARC LENGTH)—measure of its central ∠ (in °) • Reflex ∠—an ∠ whose measure is >180° • Arc Addition Postulate: • In a circle, = chords have = arcs • In a circle, = arcs have = chords

13. Inscribed angles • Inscribed angle—an angle whose vertex is on a circle, with each of the angle’s sides intersecting the circle in another point • Theorem: • An inscribed angle is equal in measure to ½ its intercepted arc • Corollaries: • Inscribed angles that intercept the same arc are equal • An angle inscribed in a semicircle is a right angle x° ½x° y° y°

14. Practice Problems! Quick Check! ½a° 90° y°

15. Lessons 5-6 & the 2 Extra lessons laura Π

16. Secant Angles Let’s start with a secant… Definition: A secant is a line that intersects a circle in two points.

17. Secant Angles Definition of Secant Angle: A secant angle is an angle whose sides are contained in two secants of a circle so that each side intersects the circle in at least one point other than the angle’s vertex.

18. Secant Angles There are two theorems involving secant angles: Theorem 64: A secant angle whose vertex is inside a circle is equal in measure to half the sum of the arcs intercepted by it and its vertical angle.

19. Secant Angles Theorem 65: A secant angle whose vertex is outside the circle is equal in measure to half the difference of its larger and smaller intercepted arcs. 100˚ 20˚ 40˚

20. Tangent Segments Definition: If a line is a tangent to a circle, then any segment of the line having the point of tangency as one of its endpoints is a tangent segment to the circle.

21. Tangent Segments Tangent Segment will come in handy when problem solving with this theorem: The Tangent Segments Theorem: The tangent segments to a circle from an external point are equal.

22. Intersecting Chords • Another theorem about two segments relating to a circle is regarding intersecting chords: The Intersecting Chords Theorem: If two chords intersect in a circle, the product of the lengths of the segments of one chord is equal to the product of the lengths of the segments of the other chord. C B AP PB CP PD P A D

23. Summary of Segments in circles While Chapter 6 is still fresh in your mind remember the following theorems that will help in problem solving... The Tangent Segments Theorem: The tangent segments to a circle from an external point are equal (Lesson 6). The Intersecting Chords Theorem: If two chords intersect in a circle, the product of the lengths of the segments of one chord is equal to the product of the lengths of the segments of the other chord (Lesson 6).

24. Summary of segments in circles You won't find these in the textbook so don't FORGET to review: The Intersecting Secant-Tangent Theorem: If a tangent and a secant intersect outside of a circle, then the square of the length of the tangent is equal to, for the secant, the product of the segment outside the circle and the entire secant (Not In Book). T A B T2= A(A+B)

25. Summary of segments in circles The Intersecting Secants Theorem: If two secants intersect outside of a circle, for each secant, the product the segment outside the circle and the entire secant are equal. (Not In Book). A B(A+B)= C(C+D) B C D

26. Summary of Segments in circles Being repetitive on purpose: All radii of a circle are equal (Lesson 1).

27. Summary of segments in circles The figure at the right illustrates a line that intersects a chord in a circle. It appears to show that line l 1) Contains the center of the circle. 2) Is perpendicular to chord AB. 3) Bisects chord AB. O B C Given that any two of these statements are true, then all three are true (Lesson 1). A l

28. Practice Problems! Angles & Arcs a˚ 90˚ a/2

29. Practice Problems! a+b/2 b˚-a˚/2 180-a˚ b-180˚ 360˚

30. Practice Problems! b-a/2 b-a/2

31. Hint Know how/when to draw additional segments in a picture that will assist with problem solving. Example: 10 10/2=5 52+52=C2 C2= 50 C= 5 rad 2 Given chord length and distance from center and must find radius, you can draw in the radius and use the Pythagorean theorem. 5 C

32. Hint Know how to express answers in appropriate form: exact form/simplest radical form or rounded correctly to a specific place

33. Summary of chapter 12 • Chapter 12 includes 6 lessons in the book and two additional lessons. There are many definitions and theorems; including formulas relating the size of angles, arcs, and segments in circles.

34. Here’s what we learned about: Circles, Radii & Chords Tangents Central Angles and Arcs Inscribed Angles Secant Angles Tangent Segments and Intersecting Chords Arcs and Angles Segments

35. Circles Diameter Radii Pi Arc Sector Degree Measure Works Cited • Worksheets and online review from Mrs. Liedell • Jacobs, Harold R. Geometry: Seeing, Doing, Understanding. Third ed. New York: W.H. Freeman, 2003. Print. Good luck on the Final everyone!