central angle semicircle arc adjacent arcs minor arc congruent arcs major arc

1. Vocabulary central angle semicircle arc adjacent arcs minor arc congruent arcs major arc

2. A central angleis an angle whose vertex is the center of a circle. An arcis an unbroken part of a circle consisting of two points called the endpoints and all the points on the circle between them.

4. Writing Math Minor arcs may be named by two points. Major arcs and semicircles must be named by three points.

5. Adjacent arcs are arcs of the same circle that intersect at exactly one point. RS and ST are adjacent arcs.

6. Within a circle or congruent circles, congruent arcs are two arcs that have the same measure. In the figure STUV.

9. Objectives Find the area of sectors. Find arc lengths.

10. Vocabulary sector of a circle segment of a circle arc length

11. The area of a sector is a fraction of the circle containing the sector. To find the area of a sector whose central angle measures m°, multiply the area of the circle by

13. Helpful Hint Write the degree symbol after m in the formula to help you remember to use degree measure not arc length.

14. A segment of a circle is a region bounded by an arc and its chord.

15. Remember! In a 30°-60°-90° triangle, the length of the leg opposite the 60° angle is √3 times the length of the shorter leg.

16. 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.

17. Objectives Find the measure of an inscribed angle. Use inscribed angles and their properties to solve problems.

18. Vocabulary inscribed angle intercepted arc subtend

19. String art often begins with pins or nails that are placed around the circumference of a circle. A long piece of string is then wound from one nail to another. The resulting pattern may include hundreds of inscribed angles.

20. An inscribed angleis an angle whose vertex is on a circle and whose sides contain chords of the circle. An intercepted arcconsists of endpoints that lie on the sides of an inscribed angle and all the points of the circle between them. A chord or arc subtendsan angle if its endpoints lie on the sides of the angle.

25. Objectives Find the measures of angles formed by lines that intersect circles. Use angle measures to solve problems.

30. Vocabulary secant segment external secant segment tangent segment

31. In 1901, divers near the Greek island of Antikythera discovered several fragments of ancient items. Using the mathematics of circles, scientists were able to calculate the diameters of the complete disks. The following theorem describes the relationship among the four segments that are formed when two chords intersect in the interior of a circle.

35. Objectives Write equations and graph circles in the coordinate plane. Use the equation and graph of a circle to solve problems.

37. 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.

38. Since the radius is , or 4, use ±4 and use the values between for x-values. Example 2A: Graphing a Circle Graph x2 + y2 = 16. Step 1 Make a table of values. Step 2 Plot the points and connect them to form a circle.