A common criticism of many children's and some adults' drawings is that certain parts are not ‘in proportion?. That means that they are either too big or too small in relation to the rest of the masterpiece. ‘In proportion? means being in the same ratio. Imagine that you have drawn a picture of the front of your house, reducing it in scale to one twentieth of its size.
If your drawing is to be ‘in proportion?, then every length detail must be drawn
of the original size. So if the door of your house is 2 m (or 200 cm) high, the door in your drawing should be 10 cm high if it is to be ‘in proportion?. In other words, if you take any length measurement from the front of your house and divide it by the corresponding measurement for your drawing, the answer should be exactly twenty.
The numerical answer that you get when you divide one measurement by another is called the ratio of their measurements.
Table 26 contains some actual body measurements along with the corresponding measurements taken from a photograph. If you have a photograph of yourself, you may care to use your own figures here.
| Actual person M (cm) | Photograph P (cm) | Person/photograph ratio M/P | |
|---|---|---|---|
| Height | 173 | 4.1 | |
| Shoulder width | 44.5 | 1.1 | |
| Arm length | 71 | 1.7 | |
| Foot length | 25 | 0.6 |
(a) Calculate the actual ratios of the measurements to the corresponding-measurements taken on the photograph and record them in the final column of the table. (Again, use a calculator.)
(b) What can you say about the size of the photograph?
(a) The four ratios are, roughly, 42.2, 40.5, 41. 8 and 41.7. So the ratio M/P is about 40 :1.
(b) The photograph is about
of life size.
Two shapes which are in proportion to each other have the same shape. In mathematics, they are said to be similar. The word ‘similar? is used rather precisely in mathematics, in contrast to how it is often used in everyday language. For example, a teacher might be unhappy that two exam scripts look so ‘similar?. Two sisters or brothers might look ‘similar?.
So, in everyday use the word means simply ‘alike in certain respects?. Just what exactly it is that makes them alike is often not made clear. In mathematics, however, the word ‘similar? means having the same shape.
The two triangles drawn in Figure 5 are said to be similar, because one is an exact scaled-up version of the other. Because they are the same shape, the corresponding (matching) angles are equal and the corresponding sides are in proportion. In this case, the sides in the larger triangle are each twice as long as those of the smaller.
Figure 5 Two similar triangles?Long descriptionAs you can see in Figure 6, when the smaller triangle is rotated and placed inside the larger one, it becomes obvious that they have the same shape.
Figure 6 Two similar triangles!Long descriptionThis redrawn form of representation allows the matching up of the sides and the angles of the two triangles so that you can observe which ones correspond with each other. Matching up these corresponding components reveals two important properties of similar shapes. First, their corresponding angles are equal (the angles marked × are equal, and so on). Second, their corresponding sides are in the same proportion—in this case, the ratio of larger to smaller is constantly two to one. This ratio is often written as 2:1 (and read ‘two to one? or ‘two is to one? or ‘the ratio two to one?).
Which of the following pairs of shapes are similar (in the mathematical sense)?
(a) Any two squares.
(b) Any two rectangles.
(c) Any two circles.
(d) Any two equilateral triangles (an equilateral triangle is one with all three sides equal in length and all three angles equal in size).
(e) Any two right-angled triangles.
(a) Yes; as you can see from the diagram below, all squares have the same shape.
(b) No; two rectangles may have the same shape, but not necessarily. They will be similar only if the ratio of length to breadth is the same for both rectangles. For example, rectangles A and B are similar, but rectangle C is not similar to either A or B.
(c) Yes; as demonstrated in the diagram below, all circles have the same shape.
(d) Yes; as you can see below, all equilateral triangles have the same shape.
(e) No; two right-angled triangles may have the same shape, but not necessarily. They will only be similar if all the corresponding angles are the same. For example, triangles A and B below are similar, but triangle C is not similar to either A or B.
An idea which is helpful in all problems on proportion or scaling is that of a scale factor. In Figures 5 and 6, the scale factor was two. In the example of the photograph (Activity 33), the scale factor was about one-fortieth.
Now for a more practical example of proportion, try scaling the ingredients of the following recipe.
The ingredients for six servings of hazelnut ice cream are given below. If you want to scale the recipe for eight servings, what is the scale factor? Complete the table for eight servings.
| Ingredients | Amounts for six servings | Amounts for eight servings |
|---|---|---|
| Toasted hazelnuts | 225 g | |
| Cornflour | 2 tablespoons | |
| Separated eggs | 2 | |
| Castor sugar | 75 g | |
| Milk | 300 ml | |
| Vanilla essence | a few drops | |
| Double cream | 300 ml |
The scale factor is 8/6 = 4/3 (or
). So multiply each number in the second column by a scale factor 4/3. In some cases, this is fairly straightforward. For example:
| Hazelnuts | 225 g × 4/3 | = 300 g |
| Castor sugar | 75 g × 4/3 | =100 g |
| Milk | 300 g × 4/3 | = 400 g |
However, others were less obvious. It is difficult to crack
eggs for example! A solution would be to use three small eggs and three tablespoons of cornflour (perhaps ensuring that the third spoon would not be quite full).
The ratio ‘amount for eight? : ‘amount for six? is generally the same for each ingredient. In general, the concept of ratio includes scale factors, and price ratios represent the scaling of prices. These ideas are all to do with things getting bigger and smaller and the sort of calculations which are used to describe such changes proportionately. Spend a few minutes now thinking about these ideas and how they are all built around the same mathematical processes of multiplication and its inverse operation, division.
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