ETIE F CICLE evision. The terms Diameter, adius, Circumference, rea of a circle should be revised along with the revision of circumference and area. ome straightforward examples should be gone over with the students. Example 1 Calculate the circumference of a circle with radius 5 cm. ns. C = πd = 3 14 x 10 = 31 4 cm Example 2 Calculate the area of a circle with diameter 20 cm. ns. = πr 2 = 3 14 x 10 x 10 = 314 cm 2 Note: some students may require further practice in finding the circumference and the area.. Length of the C of a Circle major arc rc is simply a fraction of the circumference. The fraction depends on the size of the angle at the centre. e.g. C D rc CD has 90 at centre, so CD is 90 /360 of the whole angle at the centre. Therefore, arc CD is 90 /360 of the whole outer circle, the circumference. (or 1 /4) angle at centre minor arc 20 rc has 20 at centre, so is 20 /360 of the whole angle at the centre. Therefore, arc is 20 /360 of the whole outer circle, the circumference. Mathematics upport Materials: Mathematics 1 (Int 2) taff Notes 19
The following 3 examples can be used to illustrate how to find the arc length: 5 cm 20 10 cm 8 cm C = πd C = πd C = πd = 3 14 x 10 = 3 14 x 20 = 3 14 x 16 = 31 4 cm = 62 8 cm = 50 24 cm 300 rc = 90 /360 x 31 4 cm rc = 20 /360 x 62 8 cm rc = 300 /360 x 50 24 cm = 7 85 cm = 3 49 cm = 41 9 cm Exercise 1 may now be attempted. eware 6 for extension only.. The rea of a ector ector is simply a fraction of the whole area of the circle. The fraction depends on the size of the angle at the centre. 30 angle at the centre will mean that the sector to which it belongs will have an area of 30 /360 ( 1 /12) of the area of the circle. For example Calculate the area of sector. ns. = πr 2 (watch for diameter) = 3 14 x 10 x 10 = 314 cm 2 rea of ector = 30 /360 of 314 cm 2 = 26 2 cm 2 30 10 cm Exercise 2 may now be attempted. eware 8 for extension only. Mathematics upport Materials: Mathematics 1 (Int 2) taff Notes 20
C. The elationship between Tangent and adius Note: n investigative approach is recommended for this topic. tangent to a circle is at right angles to the radius through the one point of contact. how this is tangency. This is not. n example can then be given to show how the appearance of a right angle may mean the use of ythagoras, H CH T, angles in a triangle. Example is a tangent to a circle, centre. Calculate: (a) length (b) angle (c) angle 4 cm 8 5 cm ns. the fact that is a tangent means that the angle at is 90. (a) y ythagoras Theorem (b) sin = 4 /8 5 (c) ngle 2 = 8 5 2 4 2 angle = 28 1 = 180 90 28 1 = 72 25 16 = 61 9 = 56 25 = 7 5 cm Exercise 3 may now be attempted. D. ngle in a emi-circle Note: n investigative approach could be used with this topic. Circle, centre and are same length ngle diameters and and bisect each other... in a semi-circle is a rectangle is right angled Mathematics upport Materials: Mathematics 1 (Int 2) taff Notes 21
y using this picture it should be stressed that no matter where lies on the circumference, C, the angle in the semi-circle is 90. C gain though, as with tangency, the appearance of a right angle may mean calculations involving ythagoras Theorem, H CH T etc. Example 1. Find the value of. 60 ns. 1 30 60 Make a sketch Note the isosceles triangle 2 30 Example 2. Find the values of a and b. ns. tan a = 30 /40 a = 36 9 y ythagoras Theorem b 2 = 40 2 + 30 2 = 1600 + 900 = 2500 b = 50 40 a b 30 Exercise 4 may now be attempted. Mathematics upport Materials: Mathematics 1 (Int 2) taff Notes 22
E. The Interdependence of the Centre, isector of a Chord and a erpendicular to a Chord X XY is a diameter of the circle, centre. Under reflection in the line XY, is the image of. o, XY bisects at right angles. note can be given of these statements: 1. 2. 3. Y Line from centre Line from centre at right angles Line bisecting chord at to mid-point of chord is to chord bisects the chord right angles goes through at right angles to chord the centre. If any one of the statements above is true then the other two are also true. Example ns. The radius of a circle is 10 cm. Calculate the distance from the centre to the chord which is 16 cm long. The dotted line is the required length. In right angled triangle T, using ythagoras Theorem T 2 = 10 2 8 2 = 100 64 = 36 T = 6 cm T 8 10 Note that the question could be rephrased to ask for: (i) the length of the chord, given length T and the radius. (ii) the radius, given chord and length T. Exercise 5 may now be attempted. The Checkup Exercise may now also be attempted. Mathematics upport Materials: Mathematics 1 (Int 2) taff Notes 23