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

ISBN 0-471-15045-2

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To find the angle of rotation, join O to A with a straight line and join O to A!. The angle AO A! between these two lines is the angle of rotation, which can be measured with a protractor. In this example, the angle of rotation is 90Â° counterclockwise.

Properties of Rotations

â™¦ The center of rotation is the only invariant point.

â™¦ A rotation of 180Â° about the center of rotation O is equivalent to an enlargement with a scale factor of k = â€” 1 with its center of enlargement at O.

â™¦ The object shape and the image shape are congruent shapes. We call this an isometry transformation.

â™¦ If the lines are parallel in the object, they will also be parallel in the image. We call this an affine transformation, which preserves parallelism.

â™¦ A point in the object is the same distance from the center of rotation as its image point. This means that OA = OAâ€™.

â™¦ Rotation is a direct transformation, which means that the object and image have the same sense i.e. it is a direct transformation.

Câ€™

398 RULE

References: Composite Transformations, Congruent Figures, Enlargement, Escher, Indirect Transformation, Invariant Points, Perpendicular Bisector, Rotational Symmetry.

ROTATIONAL SYMMETRY

Reference: Order of rotational symmetry.

ROUNDING

Reference: Accuracy.

ROW

Reference: Column.

RULE

References: Difference Tables, Formula, Patterns.

s

SAMPLE

A sample is a term used in statistics to refer to part of a population. If the sample is a random sample, it is chosen without bias from the whole population, and, it is hoped, has the same characteristics as the whole population. These terms are discussed more fully under the entries listed in the references.

References: Bias, Population, Random Sample.

SAMPLE SPACE

Reference: Event.

SCALE

In everyday life we use scales to measure such quantities as:

â™¦ Length, using a ruler or tape measure

â™¦ Temperature, using a thermometer

â™¦ Angles in degrees, using a protractor

â™¦ Weight, using weighing scales

â™¦ Revolutions per minute, using a revolution counter

â™¦ Volts, using a voltmeter

â™¦ Altitude, using an altimeter

A scale may be uniform or nonuniform, but in this entry we will study only uniform scales. For these scales equal distances on the scale represent equal quantities that are being measuring. The following examples illustrate how to read uniform scales.

On the voltmeter shown in figure a, 5 volts is divided up into four equal parts, so each part is 5 ^ 4 = 1.25 volts. The reading for point A is 5 + 3 x 1.25 = 8.75 volts.

399

400 SCALE DRAWING

The reading for point B is 20 + 2 x 1.25 = 22.5 volts. The reading for point C is 25 + 2.5 x 1.25 = 28.125 volts.

Figure b shows a measuring tape with 1 meter divided into five equal parts, so each part is 1 5 = 0.2 meters. The reading for point A is 3 + 2 x 0.2 = 3.4 meters.

The reading for point B is 4 + 3x0.2 = 4.6 meters. The reading for point C is 2 + 3.5 x 0.2 = 2.7 meters.

C A B

(b)

Figure c shows a measuring jug for liquids. Each 50 milliliters is divided into two equal parts, so each part is 50 2 = 25 milliliters. The reading for point A is

0 + 25 = 25 ml. The reading for point B is 150 + 25 = 175 ml. The reading for point C is 100 +0.5 x 25= 112.5 ml.

(c)

References: Meters, Milliliters.

SCALE DRAWING

Pat wanted a new house, so she visited Rob, an architect, who drew up on paper the plans for the house. The plans were obviously not the full size of the house and Rob used a scale of 1:150 for some of the drawings and a scale of 1:50 for those that required more detail. Robâ€™s plans are called the scale drawings of the house. The scale must be included with every scale drawing. Scales can be written using units, for example a scale of 1:150 can be written as 1 cm to 150 cm, or 1 cm to 1.5 meters.

SCALE DRAWING 401

Example. The scale drawing in the figure is part of Patâ€™s house. What is (a) the length of bedroom 3? (b) The length of the bathroom? (c) The area of bedroom 3?

Scale: 1:150 ,

PASSAGE ^ BATH

BED 1 BED 2

BED 3

a

Solution. A scale of 1:150 is equivalent to a scale of 1 cm to 1.5 meters.

(a) Using a ruler, we find that the length of bedroom 3 on the plan is 2.7 cm. Write

Actual length of bedroom 3 = 2.7 x 1.5 Scale is 1 cm is equivalent to 1.5 m

= 4.05 meters

(b) The measured length of the bathroom on the plan is 1.7 cm. Write

Actual length of bathroom = 1.7 x 1.5

= 2.55 meters

(c) The measured width of bedroom 3 on the plan is 2.4 cm. Write

Actual width of bedroom 3 = 2.4 x 1.5

= 3.6 meters

Then

Area of bedroom 3 = length x width = 4.05 x 3.6 = 14.58 square meters

The area of bedroom 3 is 14.58 square meters.

Reference: Ratio.

402 SCATTER DIAGRAM

SCALE FACTOR

For information on the ratio of areas and volumes of similar shapes, see the entry Similar Figures.

References: Enlargement, Ratio, Similar Figures.

SCALE OF A MAP

Reference: Ratio.

SCALENE TRIANGLE

A scalene triangle is one that has three sides of different lengths. It does not have an axis of symmetry, and its order of rotational symmetry is one. The figure shows three examples of scalene triangles.

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