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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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In mathematics when we use the word enlargement we do not necessarily mean the image is larger than the object. For an enlargement of, say, one-half, the dimensions of the image will be only half those of the object. In the work that follows it is assumed that the object shape and the image shape are in the same plane.
Example 1. In figure a, enlarge the flag F (which is the object) with a scale factor of k = 3, if the point P(2, 1) is the center of enlargement.

\ — -1 1 1 1 1 1 1 i i
1 I-.1 1 “A-r-
Solution. Draw rays from P to pass through the points of the flag. A few points on the flag are sufficient in order to be able to draw the image flag F'. Using your compasses (or dividers), step off the distance PA along the ray three times to reach the point A1. Repeat this process along the other rays and build up the image flag. Since the scale factor is k = 3, the image lengths of the flag F' will be three times the object lengths of the flag F. For example, the length of the flagpole in the object is 2 units long and the length of its image is 6 units long. For a scale factor of k = 1, the object and the image are the same shape in the same place.
Notation: We say that the flag F maps onto its image F' under an enlargement center P, and k = 3. The symbol for “maps onto” is —so we say F -> F'. In all transformations the image shape is denoted by dashes (F!) and the object shape is given without dashes (F). Similarly for points A and A1.
Sometimes the center of enlargement is inside the object shape, as in the next example.
Example 2. Draw the image of the shaded triangle ABC in figure b for an enlargement center at the origin, and k = 2.
Solution. The rays are drawn from the origin O through the points A, B, and C, respectively, and the distances stepped out with compasses to get the image points A\ B\ and C'. The image of triangle ABC is triangle A'B'C1.
There is an important rule for enlargements with regard to the ratio of image lengths to object lengths and k, which is stated here:
Image length Object length
In the previous example, when k = 2, this rule states that
o _ A’B’ _ B’C’ _ A’C’
~ ~AB~ ~ ~BC ~ ~AC
Another important rule concerns the scale factor for area. The ratio of the area of the image to the area of the object = k2. This rule is stated here:
^2 Area of image Area of object
In the previous example, when k = 2, this rule states that
Area of triangle A’B’C’
Area of triangle ABC
24 square units 6 square units = 4, which is true
For negative enlargements the image is upside down in relation to the object, and object and image are on opposite sides of the center of enlargement. The rules about ratio of lengths and about scale factor for area still apply to negative enlargements.
Example 3. Enlarge the arrow in figure c with O as the center of enlargement and with a scale factor k = — |.
Solution. It is not necessary to draw rays from the point O through all the points on the arrow, but enough need be chosen to enable the image shape to be constructed. The five points A, B, C, D, and E on the object figure should be sufficient. From the point O draw in the rays OA, OB, OC, OD, and OE, but produce them backward, since it is a negative enlargement and the image is on the opposite side of O to the object. Then halve the distance OA, since k = — |, and with compasses step off this distance of half OA along OA produced backward and mark the point A'. In a similar way mark the points B\ C', D!, and E!. Complete the drawing of the arrow. Check that the characteristics of a negative enlargement are demonstrated. The image is upside down, and the object and image are on opposite sides of the center of enlargement. Note also that the image lengths are half the object lengths.
If the object and the image are both drawn, the center of enlargement can be found by drawing the rays A! A, B'B, etc., which will all meet in the center of enlargement. The scale factor can be found using the formula
image length object length
~ ~AB~
Enlargements have the following properties:
♦ The center of enlargement is an invariant point.
♦ An enlargement with a scale factor of k = — 1 is equivalent to a rotation about the center of enlargement of 180°.
♦ Each line in the object is parallel to its own image line. For example, AB is parallel to A'B1.
♦ If lines are parallel in the object, they will also be parallel in the image. Parallelism is preserved.
♦ Angle sizes are invariant.
♦ The object and image are similar shapes.
References: Invariant Points, Similar Figures, Transformation Geometry.
Equations can be formed in order to solve some every-day problems; the process is explained under the entry Abstract. When solving an equation, only use one equals sign per line of working, and keep the equals signs in a straight line underneath each other.
Example. Solve 2x - 3 = 8.
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