Hi, I am teacher Daisy. Now, let's learn Form 2 Mathematics Chapter 5 Circles. In this chapter, you will learn 5.1 Properties of Circles, 5.2 Symmetry in Chords, and 5.3 Circumference and Area of a Circle.
5.1 Properties of Circles A circle is a curved pathway locus of a point that is equidistant from a fixed point. In geometry, a circle can be defined as a closed, two-dimensional curved shape. Balls, globes, and marbles are not considered as circles. The parts of the circle radius a straight line from the center of the circle to any point on the circumference.
Perimeter of a circle. Center, a fixed point where all points on the circumference are equidistant from it. Segment, the region enclosed by a chord and an arc.
Sector, the region enclosed by two radii and an arc. Arc, part of the circumference. Chord, a straight line that joins any two points on the circumference.
Diameter. A straight line that touches the circumference through the center of the circle. Example.
Identify the following parts of a circle. A. Cord.
B. Diameter. C. Radius. D. Circumference.
E. Sector. F. Arc. Constructing a circle.
Construct a circle when the radius or diameter is given. A. Construct a circle with a radius of 3 cm from the center O.
point in a circle. B. Construct a diameter that passes through point Q in a circle with the center O.
Construct a chord through a given point. When the length of the cord is given, C. Construct two cords of 3 cm in length, from point P on a circle. 1. Using compasses measure 3 cm on a ruler. 2. Place the sharp point of the compasses on point P. 3. Draw the arc that cuts on the circumference and label it as point A.
Construct a sector with an angle of 60 degrees at the center of a circle with a radius of 2 centimeters. 1. Draw a circle with a radius of 2 centimeters. 2. Measure 60 degrees with a protractor.
Features in a circle. The diameter of a circle is the axis of symmetry of the circle. Circle has an infinite number of axes of symmetry.
A radius which is perpendicular to the chord bisects the chord. Perpendicular bisectors of two chords meet at the center of the circle. Equal chords or chords of the same length produce arc of the same length. Equal chords are equidistant from the center of the circle. Example.
The diagram shows a circle with center O and the line MN is the chord. A. Name the axes of symmetry of this circle. B. Given OK equals 3 centimeters and NK equals 4 centimeters, calculate length of ON.
CE, name the angle that is equal to angle ONK. Solution. A, Aob, and Pocue.
B, On2, equals 42 plus 32. On equals square root of 16 plus 9, equals square root of 25, equals 5 cm. See angle Onk. The diagram shows a circle with a radius Op, that is perpendicular to the chord Mn.
A. Is the length MS equal to the length of SN? Explain. B. The radius of the circle is 10 cm and OS equals 8 cm.
Calculate the length of the chord MN. Solution. A. Yes, MS equals SN, the radius of, which is perpendicular by sex MN. B. MS equals square root of. 10 squared minus 8 squared equals square root of 100 minus 64 equals square root of 36. ms equals sn equals 6. mn equals 12 centimeters.
The diagram shows two equal chords rs and tu. Toq is a straight line passing through the center of the circle o. Given OP equals 5 centimeters and RS equals 24 centimeters. A. Calculate the length of PR.
B. Are minor arc RMS and TNU equal in length? Explain. C. Calculate the radius of the circle. Solution.
A. A radius that is perpendicular to the chord bisects the chord into two equal lengths. Length of PR equals 24 divide by 2 centimeters equals 12 centimeters. BES, chords that are equal in length, produce arc of the same length. C, OR equals square root of PR squared plus OP squared, equals square root of 12 squared plus 5 squared, equals square root of 144 plus 25. Equals square root of 169, equals 13 centimeters. Center and radius of a circle, a perpendicular bisector for any chord, will always intersect at the center of the circle.
Determine the center and radius of a circle by geometrical construction. Example, a blacksmith was asked to build a round-shaped window frame. The window is 50 cm in diameter. Three iron rods PR, US and QT that are not equal in length are used to support the window.
Calculate the length of PR. Solution Radius equals diameter divide by 2 equals 50 halves equals 25 cm OT equals square root of 25 squared minus 24 squared equals square root of 625 minus 576 equals square root of 49 equals 7 centimeters. OQ equals 31 minus 7 equals 24 centimeters. PQ equals square root of 25 squared minus 24 squared equals square root of 625 minus 576 equals square root of 49 equals 7 centimeters.
PR equals 7 plus 7 equals 14 centimeters. 5.3 Circumference and Area of a Circle. Relationship between circumference and diameter. Pi equals 3.142 or 22 over 7. Circumference divide by diameter equals pi. Circumference equals pi times diameter, equals pi d.
Diameter equals 2 times radius, thus, circumference equals pi times 2 times radius, equals 2 pi r. Formula for area of a circle. The area of a circle, equals pi r squared. Circumference, area of a circle.
Length of arc and area of sector To determine the circumference of a circle, example. Calculate the circumference of a circle. If a diameter d equals 14 centimeters, use pi equals 22 over 7. B radius r equals 21.3 centimeters, use pi equals 22 over 7. Solution a circumference equals pi d.
equals 22 over 7 times 14 equals 44 centimeters. b. Circumference equals 2 pi r equals 2 times 3.142 times 21.3 equals 133.85 centimeters.
a. Given the circumference of a circle is 88 centimeters, calculate the diameter of the circle in cm. B. Given the circumference of a circle is 36.8 cm, calculate the radius of the circle in cm, and round off the answer to two decimal places.
Solution. A. Circumference equals pi d. 88 equals 22 over 7, times d.
d equals 88, times 7 over 22. d equals 28 cm. B. circumference equals 2 pi r, 2 pi r, equals 36.8, 2 times 3.142, times r, equals 36.8, r equals 5.86 centimeters. To determine area of a circle, example, calculate the area of a circle with a diameter 10 centimeters, use pi equals 22 over 7, b.
Radius 7 cm, use pi equals 22 over 7. Solution, A, area equals pi r squared, equals 22 over 7 times, 10 over 2, square, equals 78.57 square centimeters. B, area equals pi r squared, equals 22 over 7, times 7 squared. equals 154 square centimeters.
Determining length of arc in a circle, the length of arc is proportional to the angle at the center of the circle. Length of arc over circumference equals angle at center over 360 degree. Therefore, length of arc over 2 pi r equals theta over 360 degree.
Example, The diagram shows a circle with a radius of 14 centimeters and centered at O. Calculate the length of minor arc PQ, which encloses 60 degrees at the center. Write your answer to two decimal places. Solution.
Length of arc over 2 pi r equals theta over 360 degree. Length of arc equals theta over 360 degree. Times 2 pi r, equals 60 degree, over 360 degree, times 2, times 22 over 7, times 14, equals 14.67 centimeters.
Example, the diagram shows a circle, with a radius of 21 centimeters, and centered at O. ROS is 72 degrees. Calculate the length of major arc RS. Solution. Angle of center equals 360 degree minus 72 degree equals 288 degree.
Length of arc over 2 pi r equals theta over 360 degree. Length of arc equals theta over 360 degree times 2 pi r equals 288 degree over 360 degree times 2 times 22 over 7. times 21 equals 105.6 centimeters. To determine area of a sector, the area of the sector is proportional to the area of the circle. Area of sector over area of circle equals angle at center over 360 degree.
Therefore, area of AOB over pi r squared equals theta over 360 degree. Example. The diagram shows a circle with center O and radius O.
is 21 centimeters calculate the area of the minor sector mon solution area of sector over area of circle pi r squared equals angle at center theta over 360 degree area of sector mon equals 100 degree over 360 degree times 22 over 7 times 21 squared equals 385 square millimeters. Example, given the area of the sector QOP is 18.48 square centimeters, and the radius is 12 centimeters. Calculate the value of theta.
Solution, area of sector, over area of circle, pi r squared, equals angle at center, theta. over 360 degree, theta over 360 degree, equals 18.48 divide by, 22 over 7, times 12 squared, theta equals 18.48 divide by, 22 over 7, times 12, times 12, times 360 degree, theta equals 14.7 degree. Solving problems, matchless Bandarya Malaka Bersahara intends to build a rectangular recreational park with a length of 63 meters and a width of 58 meters. At every corner of the park, a quadrant with radius 7 meters will be planted with flowers. A circular shaped fish pond with a diameter of 28 meters will be built in the middle of the park.
The remaining areas will be planted with grass. Calculate the area covered with grass. Solution, Recreational Park Area, equals 58 times 63, equals 3,654 square meters. Flower Area, equals 4 times 1 quarter, times pi r squared, equals 22 over 7 times 7 squared, equals 154 square meters.
Fish pond area equals pi r squared, equals 22 over 7 times 14 squared, equals 616 square meters. Area covered with grass is, 3,654 square meters minus 154 square meters, minus 616 square meters, equals 2,884 square meters. The concept map for Form 2 Chapter 5 is as below. If you find this video helpful, don't forget to like, share and subscribe our channel.
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