Mth 076: Applied Geometry (Individualized Sections) MODULE FOUR STUDY GUIDE


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1 Mth 076: pplied Geometry (Individualized Sections) MODULE FOUR STUDY GUIDE INTRODUTION TO GEOMETRY Pick up Geometric Formula Sheet (This sheet may be used while testing) ssignment Eleven: Problems Involving ircumference and rea of ircles. Read pages 51 & 5 in your textbook. Study examples 13 odd on page 5.. Work examples 4 even on page 5 and check your results.. ssignment Eleven: Do problems, 4, 88 even in Exercise 4, Objective, pages 53, 54. ssignment Twelve: Problems Involving Definitions and Properties of ircles. Read and study pages 13 in the Study Guide.. ssignment Twelve: Do problems 14 on pages 3 & 4 in the Study Guide. ssignment Thirteen: Problems Involving Length of rc and rea of a Sector. Read pages 5 & 6 in the Study Guide. Study examples 14 in the Study Guide.. ssignment Thirteen: Do problems 110 on pages 68 in the Study Guide. Review ssignment: Do the module four review assignment available in the math learning center and submit it to an instructor for grading. Module Test: fter successfully completing the three assignments and the review assignment, ask an instructor for a module four test. 1
2 ircles: Some More Definitions and Properties hord: line segment having its end points on the circle. Segment is a chord Tangent line: line that touches the circle at exactly one point. L 1 L 1 is a tangent line. Point is the point of tangency. Secant line: line that passes through two points of the circle. L L is a secant line. entral ngle of a ircle: n angle with its vertex at the center of the circle. is a central angle, given that point is the center of the circle. rc of a ircle: The part of the circle between and containing two specified points on the circle. There are two such arcs on a circle, the minor arc and the major arc. The portion of the circle from point to point is the minor arc. The portion of the circle from point through point to point is the major arc. Note that a minor arc is indicated with two points and a major arc is indicated with three points.
3 Intercepted rc: The arc of a circle between the two rays of an angle. intercepts arc. Measure of the rc of a ircle: n arc is measured by its central angle. The measure of an arc of a circle and the central angle that intercepts the arc are equal. The measure of the minor arc is 80 o since the D 80 o measure of the central angle is 80 o. The measure of the major arc D is 80 o since o  80 o = 80 o. Inscribed ngle: n angle is inscribed in an arc if the sides of the angle contain the end points of the arc and the vertex of the angle is a point on the arc. (Not an end point) is an inscribed angle since its vertex is on arc. intercepts arc. Measure of an inscribed angle: The measure of an inscribed angle is onehalf of the measure of its intercepted arc. Given: rc measures 150 o Since the measure of arc is 150 o the measure of is 75 o. 3
4 Since the measure of is 90 o, the measure of arc is o. This means is a diameter of the circle. Tangent Line and Radius of a ircle: tangent line to a circle is perpendicular to the radius of a circle drawn to the point of tangency. D, so and D are right angles D ssignment #1 From the figure at the right, identify the following: Point E is the center of the circle. 1. a) secant line b) tangent line c) central angle d) n inscribed angle e) minor arc D I E F G f) major arc g) Two perpendicular lines h) chord H 4
5 . Given D = 50 o, D is a diameter. Find: a) Measure of arc b) Measure of D c) Measure of D d) Measure of arc D e) Measure of DE f) Measure of ED D E 3. Given the measure of D = 35 o, is the center of the circle. Find the measure of arc E. D E 4. Given the measure of arc D = 00 o Find the measure of E D E 5
6 Length of rc of a ircle We have found that the measure of an arc is equal to the measure of its central angle and the measure of entire circle is o. The ratio of the length of arc of a circle to the circumference of a circle is the same as the ratio of the measure of the central angle of the arc to o. Symbolically, this is given by L n where L represents the length Π r = of arc, n is the measure of the central angle of the arc in degrees and r is the radius of the circle. Solving for L gives: L =. Example 1: Find the length of arc given a radius of 5 inches and a central angle of 30 o. Solution: L =, r = 5, n = 30. Substituting for the appropriate variables gives: L = ( 30)( Π)( 5). Simplifying the right side of the equation gives L = 5Π inches. Thus 6 L.6 inches. Example : Find the measure of the central angle given the length of arc is cm. and the radius is 10 cm. Solution: L =, L =, r = 10. Substituting for the appropriate variables gives: nπ( 10) =. Solving for n gives ( )( ) 396 = n or n =. Thus n 16 o. ( 10)( Π) Π rea of a Sector of a ircle The sector of a circle is the region bounded by an arc and two radii of the circle. The ratio of the area of a sector to the area of the circle is equal to the ratio of the measure of the central angle of the arc created by the two radii of the sector to o. 6
7 Symbolically, this is given by n =, where represents the area of the sector, n Πr represents the measure of the central angle of the arc in degrees and r represents the length of the radius of the circle. Solving for gives =. Example #3: Find the area of a sector given its radius is 10 inches and the measure of its central angle is 130 o. Solution: Since =, substituting 130 for n and 10 for r yields = 130Π( 10). Evaluating the right side of the equation gives = 35 square inches or approximately 9 Π square inches. Example #4: Find the radius of a sector given its central angle is 75 o and the area of the sector is 35 square centimeters. Solution: Since =, substituting 75 for n and 35 for yields 35 = 75Π( r). y multiplying each side of the equation by gives (35)() = 75 Πr. Dividing each side of the equation by 75 gives ( 35)( ) Π = r. Taking the square root of each 75Π side of the equation solves for r yielding ( 35)( ) = r 75Π or r is approximately centimeters. ssignment #13 1. Determine the lengths of arc from the given information: a) r = 1 in. ; central angle = 5 o b) d = 5 m. ; central angle = 130 o. Find the measure of the central angle of an arc given the length of the arc is 8 in. and the radius measures 17 in. 3. Find the radius of an arc given the length of the arc is 34 cm. and the central angle measures 95 o. 7
8 4. Find the perimeter of the figure to the right. The line segments are tangent to the arc. 4 ft ft. 5. Find the length of the pulley belt needed to reach around the two pulleys. 6 in. 0 in. 150 o 4 in. 0 in. 6. Determine the area of the sectors from the given information: a) r = 1 in. ; central angle = 5 o b) d = 5 m. ; central angle = 130 o 7. Find the central angle of the sector given the area of the sector is.5 square centimeters and the radius is 4. centimeters. 8. Find the radius of the sector given the area of the sector is 7.3 square miles and the central angle measures 140 o. 9. Find the area of sector created by the central angle, given =.3 feet. 60 o.3 ft. 8
9 10. Find the area of the following region. The region was created by removing circular sectors from a regular hexagon. The dotted lines are on the hexagon and intersect at the centers of the circles. 4 ft. 6 ft. 4 ft. 9
radii: AP, PR, PB diameter: AB chords: AB, CD, AF secant: AG or AG tangent: semicircles: ACB, ARB minor arcs: AC, AR, RD, BC,
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