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The A to Z of mathematics a basic guide - Sidebotham T.H.

Sidebotham T.H. The A to Z of mathematics a basic guide - Wiley publishing , 2002. - 489 p.
ISBN 0-471-15045-2
Download (direct link): theatozofmath2002.pdf
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Solution. The angle required is angle x in figure b:
x = 34° Angles on the same arc are equal The coach’s shooting angle is 34°, the same as William’s.
Example. In figure c, AB is parallel to CD, indicated by the arrows. The straight lines AC and BD intersect at point E. Prove that triangle ABE is an isosceles triangle.
Solution. Write
Angle ACD = angle ABD Angles on the same arc are equal
Angle ACD = angle BAC Alternate angles are equal
Angle ABD = angle BAC Both angles are equal to angle ACD
Therefore, triangle ABE is isosceles, because its base angles are equal.
References: Alternate Angles, Circle Geometry Theorems, Geometry Theorems, Isosceles Triangle, Subtend.
This is the region between two concentric circles, and is drawn shaded in figure a. Another name for an annulus is a ring. If the radii of the circles are R and r, then the area of the annulus is given as follows:
A = TtR1 — 7tr2 Formula for the area of a circle is A = ttR2
A = tt(R2 — r2) Factoring, since tt is a common factor
A = tt(R — r)(R f r) Factoring, using the difference of two squares
36 ARC
Example. Find the volume of plastic required to make a plastic pipe (figure b) which is 20 cm long and whose inner and outer radii of the circular ends are 3 and 4 centimeters, respectively.
Solution. Write
Volume = area of annulus x length
Volume = tt(R — r) (R + r) x length Volume = Tt(4 - 3) (4 + 3) x 20
Volume = 439.8 cm3 (1 dp)
References: Concentric Circles, Factoring, Ms
Reference: Accuracy.
An arc is a section, or part, of a curve. It can also be a section of a line graph. In the figure the arc AB is part of a circle.
When the arc of a circle is less than a semicircle, it is called a minor arc, and when the arc is greater than a semicircle, it is called a major arc. In the figure, the arc AB of the circle is a minor arc because it is less than a semicircle.
Volume of a prism = area of cross section x length
Using formula for area of annulus
Substituting R = 4, r = 3, and length = 20
Using calculator
References: Line Graph, Networks.
The length of an arc of a circle can be calculated as in the following example.
Example. Amanda is schooling up her horse George for a show. He is trotting in a circle at the end of a rope held by Amanda, who is at point A in the figure. The length of the rope is 10 meters.
(a) As Amanda turns through an angle of 120° find the distance trotted by George along the arc of the circle from G\ to G2.
(b) If George trots a distance of 20 meters, through what angle has the rope turned?
Solution, (a) The circumference C of the whole circle is given by the formula C = 7t x diameter
C = 7t x 20 Substituting 20 meters for the diameter of the circle
C = 2Q7T
The arc length trotted by George is a fraction of C, and this fraction is 120°/ 360°, where 120° is the angle turned through by the rope, and a full turn is 360°.
Arc length = x 20jr Multiplying the fraction of the full turn by the
= 20.9 (1 dp)
The distance trotted along the arc of the circle by George is 20.9 meters.
(b) Suppose the angle turned through by the rope is x. The equation is
x C = 20
360 x 20
360 x 20
jc = 114.6° (1 dp)
Making x the subject of the formula
Substituting C = 20jr from earlier Using the calculator
The rope turns through an angle of 114.6°. For an alternative method using radians refer to the entry Radian.
References: Arc, Circumference, Radian.
The area of a surface is a measure of the two-dimensional space it occupies. It is measured in square units, which is written as units2. We can find the area of a shape, say a rectangle, by counting the number of squares that its surface occupies. The area of the rectangle in figure a is found by counting the number of square centimeters it occupies. Square centimeters are abbreviated cm2. The area of this rectangle is 12 cm2. Alternatively, the length of the rectangle, which is 4 cm, and the width, which is 3 cm, can be measured and the area calculated using the formula
Area = length x width A = 4 x 3 = 12
The area of this rectangle is 12 cm2.
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For some shapes, like circles, it is difficult to find the area accurately by counting squares, because the process would involve piecing together small parts of the circle to make up whole squares, like a jigsaw. The only satisfactory method is to use a formula to find its area.
Example. Calculate the area of the circle in figure b.
Solution. We need to know the radius of the circle, which is R = 1.5 cm. We write Area = 7t x 1.52 Formula for the area of a circle is Area = tt x R2
Area = 7.07 (to 2 dp) Using a calculator The area of the circle is 7.07 cm 2.
The units commonly used for area are as follows:
♦ Square millimeters, mm2.
♦ Square centimeters, cm2.
♦ Square meters, m2.
♦ Hectares, ha.
♦ Square kilometers, km2.
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