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Area of semicircle = R2 Area of a whole circle = ttR2
= | x it x 13.52 The radius of a circle is half the diameter = 286 to nearest whole number
The area of the fan is 286 cm2.
The perimeter of the fan is half the circumference of the circle + the diameter of the circle. Write
Perimeter of fan = \nD + 27 Circumference of circle = nD
= | x it x 27 + 27 = 69 to nearest whole number The perimeter of the fan is 69 cm.
References: Area, Chord, Circle, Radius, Sector of a Circle.
References: Indirect Transformation, Reflection.
A septagon is a polygon with seven sides. It is more often called a heptagon. See the entry Polygon for the angle sum of the septagon. Some septagons are drawn in figure a; the last one is a regular septagon.
The properties of the regular septagon are as follows: Its seven sides are of equal length, and its seven angles are all equal to 128| degrees, or 128.6°, to 1 dp. It has seven axes of symmetry, and the order of rotational symmetry is seven.
The regular septagon is made up of seven isosceles triangles. An enlarged view of one of these triangles is shown in figure b. The sizes of the angles have been rounded to 1 dp.
References: Axis of Symmetry, Heptagon, Isosceles Triangle, Polygon, Rotational Symmetry, Rounding.
References: Difference Tables, Patterns.
References: Système International d’Unités, CGS System of Units.
A side is one of the line segments that form a polygon. For example, in the figure, one side of the triangle ABC is the line segment AS. Alternatively, a side can refer to one of the faces of a polyhedron. For example, the shaded region of the cuboid in the figure is called a side of the cuboid. The cuboid has six sides.
408 SIMILAR FIGURES
References: Cuboid, Line segment, Polygon, Polyhedron.
SIEVE OF ERATOSTHENES
References: Eratosthenes’ Sieve, Prime Number.
When one plane figure is an enlargement of another plane figure we say the two figures are similar. If two figures are similar, they are the same shape; for example, both figures may be triangles, or both figures may be hexagons. Similar figures have the following properties:
♦ The angles in one figure are equal to the corresponding angles in the other figure.
♦ Corresponding sides in the two figures are in the same ratio.
Example 1. David and his son William have tee shirts of the same design, but different sizes. The patterns that William’s mom, Jane, needs for making the shirts are shown in figure a. All the lengths on David’s shirt are two times the lengths on
SIMILAR FIGURES 409
William’s shirt. Therefore the shirt patterns are similar figures, except for the motifs on the fronts.
The lengths of corresponding sides of the patterns are in the same ratio. If the waist length AB on William’s pattern is 30 cm and the waist length XY on David’s pattern is 60 cm, we can write down the equal ratios:
Length of AB 30
Length of XF “ 60
This ratio of | can be called the scale factor of the enlargement of David’s pattern to William’s. This means that the lengths on William’s pattern are one-half of the corresponding lengths on David’s pattern.
On William’s shirt pattern the length of AC = 22.5 cm, so the corresponding length on David’s shirt pattern is XZ, which is 2 x 22.5 = 45 cm.
The ratio of the areas of the two shirts is the square of the ratio of the lengths:
Ratio of areas = (|)2 Squaring the scale factor for lengths Ratio of areas = \
This ratio is also true for areas of corresponding parts of the two patterns. For example, the ratio of the areas of the two semicircular necklines which have been cut out is also 1:4.
Suppose we are comparing two solid shapes where one is an enlargement of the other, like two bottles. When one solid shape is an enlargement of the other we say that the two solids are similar shapes.
Example 2. The two perfume bottles in figure b are similar shapes. The diameter of the base of the smaller bottle is one-third the diameter of the base of the larger bottle. If the smaller bottle holds 10 milliliters, how much does the larger bottle hold?
410 SIMULTANEOUS EQUATIONS
Solution. The ratio of the volumes (or capacities) of the two bottles is equal to the cube of the ratio of the lengths, which is the scale factor. Write
The larger bottle holds 27 times the volume (or capacity) of the smaller bottle, which holds 10 ml. Write
The capacity of the larger bottle is 270 milliliters, or 0.27 liters.
References: Area, Capacity, Corresponding Angles, Corresponding Sides, Cross Multiply, Cube, Enlargement, Liters, Milliliters, Ratio, Reflection, Scale Factor, Square, Volume.
References: Compound Interest, Percentage.
Simultaneous equations are two or more equations that are to be solved at the same time. The simultaneous equations we deal with here are sets of two equations in two variables, which represent two unknown quantities. A solution to the simultaneous equations is a value for each of the two variables that make both of the equations true at the same time hence the word simultaneous. An example best illustrates the concept.