Arc Measures
Arc measures refer to the size of an arc in a circle, typically measured in degrees. The measure of an arc is equal to the measure of its corresponding central angle. This concept is important in geometry and trigonometry for understanding angles and their relationship to circles.
3 Key excerpts on "Arc Measures"
- Rebecca Dayton(Author)
- 2013(Publication Date)
- Research & Education Association(Publisher)
...Chapter 9 Circles Your Goals for Chapter 9 1. You should be able to identify the parts of a circle. 2. You should be able to determine the measure of arcs and the angles created in a circle. 3. You should be able to solve problems using circumference, arc length, and areas of circles and sectors. 4. You should be able identify the center and radius of a circle and graph the circle on a coordinate plane from an equation in center-radius form. 5. You should be able to write an equation of a circle given the center and radius. Standards The following standards are assessed on Florida’s Geometry End-of-Course exam either directly or indirectly: MA.912.G.6.2: (Low) Define and identify: circumference, radius, diameter, arc, arc length, chord, secant, tangent and concentric circles. MA.912.G.6.4: (Moderate) Determine and use measures of arcs and related angles (central, inscribed, and intersections of chords, secants and tangents). MA.912.G.6.5: (High) Solve real-world problems using measures of circumference, arc length, and areas of circles and sectors. MA.912.G.6.6: (Moderate) Given the center and the radius, find the equation of a circle in the coordinate plane or given the equation of a circle in center-radius form, state the center and the radius of the circle. MA.912.G.6.7: (Moderate) Given the equation of a circle in center-radius form or given the center and the radius of a circle, sketch the graph of the circle. Lines and Segments A radius is a segment whose endpoints are the center and any point on the circle. A chord is a segment whose endpoints are on the circle. A diameter is a segment whose endpoints are on the circle and passes through the center. A diameter is the longest chord of the circle. A secant is a line that intersects the circle at two points. A tangent is a line that intersects the circle at exactly one point. Example: Identify each of the following in the given circle. A. chord B. secant C...
- Lawrence S. Leff, Christina Pawlowski-Polanish(Authors)
- 2021(Publication Date)
- Barrons Educational Services(Publisher)
...The notation 28°30′ is read as “28 degrees, 30 minutes.” Since 60 minutes is equivalent to 1 degree, dividing 30 minutes by 60 changes 30 minutes to a fractional part of a degree. Thus: TO CHANGE MINUTES TO DEGREES MEASURING ANGLES IN RADIANS In Figure 9.1, angle θ cuts off an arc of circle O that has the same length as the radius of the circle. Angle θ measures 1 radian. Unlike degrees, radians are real numbers. FIGURE 9.1 Defining a radian in a circle with radius r Since the total number of radii that can be marked off along the circumference of the circle is 2π, the radian measure of a circle is 2π. The degree measure of a circle is 360°. Hence, 2π radians = 360°, so π radians = 180°. This relationship provides the conversion factor for changing from one unit of angle measure to the other. RADIAN AND DEGREE CONVERSIONS An angle of 1 radian is the angle at the center of a circle whose sides cut off an arc on the circle that has the same length as the radius of the circle. • To convert from degrees to radians, multiply the number of degrees by. • To convert from radians to degrees, multiply the number of radians by. EXAMPLES To convert 60° to radian measure, multiply it by : To convert radians to degrees, multiply it by : CONVERSIONS WORTH REMEMBERING •, and multiples such as. •, and 360° = 2π. Lesson 9-2: Right-Triangle Trigonometry KEY IDEAS The Pythagorean theorem relates the measures of the three sides of a right triangle. A trigonometric ratio relates the measures of two sides and one of the acute angles of a right triangle. THE THREE BASIC TRIGONOMETRIC RATIOS In Figure 9.2, Δ ABC, Δ ADE, and Δ AFG each contain a right angle and they have ∠ A in common. When the ratio of the length of the leg opposite ∠ A to the length of the hypotenuse is computed, it is found to be the same for each right triangle: This constant ratio is called the sine of ∠ A...
eBook - ePub- Douglas K. Brumbaugh, Peggy L. Moch, MaryE Wilkinson(Authors)
- 2004(Publication Date)
- Routledge(Publisher)
...Carpenters use an instrument called a square to verify right angles and we can express the same concept by using the terms perpendicular and 90°. The definitions of acute and obtuse depend on our knowledge of right angles. A common error is to read the wrong scale when using a protractor. To overcome this tendency, always decide whether an angle appears to be acute, right, obtuse, or straight before you measure. The angle in Fig. 5.26 appears to be an acute angle, but the ray goes through two rules, 150° as well as 30°, and we must make a decision about which scale to use. In this case, a measure of 150° does not make sense for our acute angle. Notice that the correct scale is the one that counts up as we follow the rotation from the first leg, at 0°, toward the second leg, at 30°. Your Turn 40. How many degrees are in one full rotation? 41. Do the lengths of the legs of an angle have any meaning in degree measure? 42. Is there a special name for an angle of 180°? 43. Identify each named angle in Fig. 5.27 as less than, exactly, or greater than 90°, then measure the angles using a pro-tractor. Fig. 5.27. Angles are created if two lines, rays, or segments intersect. An application of angle measure occurs when parallel lines are cut by a transversal (a line in the same plane as the parallel lines that intersects each of the lines). Figure 5.28 shows that eight angles are formed when a transversal cuts through two parallel lines. You can determine the measure of each of those eight angles after making only one measurement. Suppose you measure Angle 1 and find that it is 35°, then Angles 4, 5, and 8 also measure 35°. Angles 1 and 4 are formed by the same pair of lines, share a vertex, and are directly opposite one another across that vertex—they are called vertical angles and they have the same measure...
