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

ISBN 0-471-15045-2

**Download**(direct link)

**:**

**86**> 87 88 89 90 91 92 .. 126 >> Next

314

PARADOX 315

Another application of the parabolic curve is in the telescope and the radiotelescope, which is used in astronomy. A parabolic “dish” is designed to receive parallel radio waves from distant planets and stars and to concentrate those waves at its focus.

References: Conic Sections, Quadratic Graphs.

PARADOX

A paradox is a statement that comes from apparently logical reasoning, but that, on closer inspection, gives an obviously absurd result. Perhaps the most famous paradox is one that was offered by Zeno, a Greek philosopher who lived about 450 BC. He described a race between Achilles and a tortoise. Achilles can run 10 times as fast as the tortoise, so he gave the tortoise a 100-pace head start. At the beginning of the race Achilles is at the point Ao and the tortoise at the point To and the distance between the two points is 100 paces. The race starts.

When Achilles has run to the point To, the starting position of the tortoise, the tortoise has moved to 7i, a distance of 10 paces (see figure). When Achilles arrives at Ti, the tortoise has moved 1 pace to T}. When Achilles arrives at Ti the tortoise has moved 0.1 paces to F3. When Achilles arrives at F3 the tortoise has moved 0.01 paces to F4, and so on. In this way, if the reasoning is continued, the tortoise will always be ahead of Achilles (albeit by an ever-decreasing distance) and so can never be caught. Common sense contradicts this conclusion. We know that the winner of the race, if it goes on long enough, will be Achilles, because he can run faster than the tortoise. But it is difficult to find a flaw in Zeno’s argument, and so we have a paradox that baffled mathematicians for centuries.

100 paces 10 paces 1 pace

Suppose the distances run by Achilles are written down as a sum, given by D:

D = 100 + 10+ 1 -1-0.1 -1- 0.01 + 0.001 + 0.0001 + ... and so on = 111+ 0.1 + 0.01 + 0.001 +0.0001 + ...

316 PARALLELOGRAM

Which is lll.i, written as a recurring decimal.

5 = 0-1

The distance run by Achilles when he catches up with the tortoise is 1111 paces. Reference: Decimal.

PARALLEL

Reference: Gradient.

PARALLELOGRAM

A parallelogram is a quadrilateral, which means a four-sidedpolygon, with its opposite sides parallel and equal in length. In addition, its opposite angles are the same size. The angles a and b in figure a are called cointerior angles and their sum is 180°, which means a + b = 180°.

Each diagonal divides the parallelogram into two congruent triangles (see figure b). The diagonals of the parallelogram bisect each other. It has no axes of symmetry. It has rotational symmetry of order two.

The shapes rhombus, rectangle, and square are classed as special kinds of parallelograms, but have their own individual properties. More information about these shapes is given under their separate entries.

To find the area of a parallelogram we need to know the lengths of its base (b) and its altitude, that is, its perpendicular height (h). It is not possible to find the area of a parallelogram if we are only given the lengths of its four sides. To find the area of the parallelogram in figure c write

(a)

(b)

4 cm

5 cm

(c)

PARALLELOGRAM 317

Area of parallelogram = base x perpendicular height = 5x4 = 20

The area of the parallelogram is 20 cm2.

It is not possible to find the perimeter of a parallelogram if we are only given the lengths of its base and altitude.

In figure d the rectangle and the parallelogram have the same area because they have the same base and the same perpendicular height, but the perimeter of the parallelogram is greater than that of the rectangle.

Example. The handrail on a set of steps on a hillside is shown in figure e. The angle marked x is 60°. It is planned to cover each panel (a panel is shaded) with “safety” plastic. Using the measurements marked on the figure, find the area of one panel.

Solution. A panel is a parallelogram, and to find the area of a parallelogram we need the two dimensions, base (b) and perpendicular height (h). These dimensions are shown on a panel drawn in the figure.

318 PASCAL’S TRIANGLE

The base is b = 80 cm and the height (h) is found using trigonometry in the small right-angled triangle. Write

— = sin 60° Sine = °PP°site

50 hypotenuse

h = 50 x sin 60° h =43.3 cm (to 3 sf)

Therefore

Area of parallelogram panel = b x h

= 80 x 43.3 = 3460 cm2 (to 3 sf)

The area of one panel is 3460 cm2.

References: Area, Base (geometry), Cointerior Angles, Constructions, Perimeter, Polygon, Quadrilateral, Rectangle, Rhombus, Square, Symmetry, Trigonometry.

PARENTHESES

This is another name for round brackets (), which are used in algebra to enclose an expression so that it can be treated as a single quantity. Once the brackets have served their purpose, they are often removed by expanding them.

References: Brackets, Expanding Brackets, Rectangle.

PASCAL’S TRIANGLE

The use of Pascal’s triangle and its symmetry patterns are explained in the entry Coefficient. Pascal’s triangle contains natural numbers, triangle numbers, and powers of 2.

**86**> 87 88 89 90 91 92 .. 126 >> Next