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

ISBN 0-471-15045-2

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Example 2. If the same cart wheel makes 400 revolutions per minute, find the speed of the cart in meters per minute.

390 rhombus

Solution. Using the answer from the previous example, we know that the cart travels it x 20 cm in one revolution, if we use 62.8 cm instead, the result will not be as accurate. Write

400 x 7T x 20 = 25,133 cm Distance cart travels in 1 minute, to nearest

whole number

This means that the cart travels 25,133 cm in 1 minute. Now write

25,133 100 = 251.33 meters 100 cm = 1 meter

The speed of the cart is 251.33 meters/minute, or 15.1 kilometers/hour, to 1 dp. Multiplying meters/minute by 0.06 changes it to kilometers/hour.

References: Circumference of Circle, Right Angle.

RHOMBUS

A rhombus is a parallelogram with the extra property that all of its four sides are the same length. In everyday life it is sometimes called a diamond. Another property of the rhombus is that its two diagonals bisect each other at right angles, but the two diagonals are not the same length (see figure a).

The rhombus has a rotational symmetry order of two, like the parallelogram, but unlike the parallelogram, the rhombus has two axes of symmetry, which are its two diagonals. These axes of symmetry can be clearly seen if the rhombus is drawn on a square grid, as shown in figure b.

(b)

RHOMBUS

391

If the lengths of the diagonals of a rhombus are a and b, then the area of the rhombus is given by

Area = | ab

The lengths of the diagonals of the rhombus that are drawn on the grid in figure b are a = 6 and b = 4. Therefore

Area of this rhombus = | x 6 x 4 Using the formula for the area of a

rhombus

= 12

The area of this rhombus is 12 square units.

The area of a rhombus is also equal to base x height, like the parallelogram.

S ince the rhombus is made up of four congruent right-angled triangles, it is possible to use the Pythagoras Theorem in one of these triangles, as in the following example.

Example. Ron is having a diamond-shaped stained glass window put into a wall of the church to give light to a particularly dark corner. He wants the width of the window to be 3 meters and the height to be 4 meters, as shown in his preliminary sketch in the figure. How long will the total length of the window frame be?

Solution. A diamond is taken to mean a rhombus, and the lengths of the diagonals are 4 and 3 meters, respectively. The diagonals divide the rhombus into four right-angled triangles that are all congruent. One of these triangles is drawn shaded in the figure. The lengths of the sides are 2 and 1.5 meters, respectively. Suppose we let the length of the hypotenuse of this triangle be x meters. Write

x2 = 22 + 1.52 Pythagoras’ theorem

x2 = 4 + 2.25 Squaring the numbers

x2 = 6.25

x = V6.25 Taking square roots

x = 2.5

392 RIGHT PYRAMID

Then

Total length = 4 x 2.5 The total length of four sides = 4 x length of one side = 10

The total length of the window frame is 10 meters.

References: Congruent Figures, Diagonal, Hexagon, Hypotenuse, Parallelogram, Regular Polygon, Right Angle, Rotational Symmetry, Symmetry.

RIGHT ANGLE

Reference: Acute Angle.

RIGHT PRISM

This is a prism in which the sides are at right angles to the uniform cross section. The figure shows a right triangular prism in which the uniform cross section (drawn shaded) is at right angles to each of the three sides. Another example of a right prism is a cylinder.

References: Cross Section, Cylinder, Prism.

RIGHT PYRAMID

This is a pyramid that has an axis of symmetry at right angles to its base. The figure shows a right square-based pyramid in which the axis of symmetry VO is perpendicular to the base. Another example of a right pyramid is a cone.

V

References: Cone, Pyramid, Right Prism.

ROMAN NUMERALS 393

RIGHT TRIANGLE

This is a triangle in which one of the angles is a right angle, and is used frequently in mathematics, especially in trigonometry and the Pythagoras Theorem. The longest side of a right triangle is called the hypotenuse. Another name for a right triangle is a right-angled triangle.

References: Pythagoras’ Theorem, Trigonometry.

ROMAN NUMERALS

The number system that we use today enables us to express all numbers by means of the 10 symbols {0,1,2,3,4,5,6,7,8,9} with each symbol varying in value according to the position or place it occupies. For example, in the number 32 the three stands for 3x10, and in the number 324 the three stands for 3x 100.

The mathematicians of ancient Rome did not use that principle of place value, and consequently their number system was clumsy and difficult to use. The Roman numerals and their equivalents are as follows:

♦ I is 1

♦ Vis 5

♦ X is 10

♦ L is 50

♦ C is 100

♦ D is 500

♦ M is 1000

Other numbers are formed from these numerals. Roman numerals have a fixed value wherever they appear, unlike our present system. When two numerals appear side by side we add them. Some examples are

MM = 1000 + 1000 = 2000 CCC = 100 + 100 + 100 = 300 DC = 500 + 100 = 600

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