# Enlarging and reducing plane figures and equivalent areas

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Beginning at R. draw the sides of the larger figure parallel to the sides of the original smaller figure.

This construction works equally well for reducing the size of a plane figure. Fig. 7/2 shows an irregular hexagon reduced to 4/9 its original size.

These constructions are practical only if the figure which has to be enlarged or reduced has straight sides. If the outline is irregular, a different approach is needed. Fig. 7/3 shows the face of a down in two sizes, one twice that of the other. The change in size is determined by the two grids. A grid of known size is drawn over the first face and then another grid, similar to the first and at the required scale, is drawn alongside. Both grids are marked off. from A to J and from 1 to 6 in this

Definition

Similar figures are figures that have the same shape but may be different in size.

Consttuctions

To construct a figure, similar to another figure, having sides 7/6 the length of the given figure.

Three examples, using the same basic method, are shown in Fig. 7/1.

Select a point P. sometimes called the centre of similitude. in one of the positions shown.

From P draw lines through all the corners of the figure. Extend the length of one of the knee from P to a comer, sey Pa in the ratio 7:6. The new length is PR.

Urheberrechtlich geschütztes Materi;

It It sometimes necessary to enlarge or reduce a plane figure in one direction only. In this case, although the dimentiont are changed, the proportions remain the same Fig. 7/4 shows a simple example of this. The figure has overall dimensions of 4 cm x 4 cm. The enlarged version retains the original proportions but now measures 6 cm >: 4cm.

First produce CA and BA. Mark off the new dimensions along CA and BA produced. This gives AB' and AC'.

Draw the square AB'XB end the rectangle A CYC' and draw the diagonals AX and AY.

From points along the periphery of the onginal plane figure (in this case 1 to 10). draw lines horizontally and vertically to and from the diegonals to intersect in 1'. 2'. 3'. etc. Points 1' to 10* give the new profile

Fig. 7/5 shows how a figure can be reduced on one side and enlarged on the other. A basic 50 mm * 50 mm shape has been changed proportionally into an 80 mm * 30 mm figure. Although this figure is more complicated than Fig 7/4. with a conetponding increase in the number of points plotted, the basic construction is the same.

There is very little practical application of this type of construction these days. When plasterers produced flamboyant ceilings with complicated cornices, and carpenters had to make complex architraves and mouldings.

GIVEN FIGURE

this type of construction was often employed However.

it is still a good exercise in plane geometry and does occasionally find an application.

The enlarged or reduced figures produced in Figs 7/4 and 7/5 are mirror images of the original figures. Usually this does not matter, particularly if the figure is for a template; it just hes to be turned over. However, if it does matter, a construction similar to that used in Fig. 7/6 must be used. In this case, a basic 60 mm * 40 mm shape has been changed into a 30 mm * 20 mm shape.

A'B and A C' are drawn parallel to AB and AC and marked off 20 mm and 30 mm long respectively

AA' and BB'. AA' and CC' ere produced to meet in Q and P respectively.

The curved part of the figure is divided into at many partt at it necessary to produce an accurate copy.

The rest of the construction should be self-explanatory.

The transfer of the markings along A'B' and A'C' on the original figure to the required figure is made easier by the use of • 'trammel'; this is a rather pompous trtle for a piece of paper with a straight edge. If you lay this piece of paper along A'C' on the given figure end mark off A', C and all the relevant points in between, you can line up the paper with A' and C on the required figure end transfer the points between A' and C' onto the required figure. The same thing can be done for A'B'.

REQUIRED FIGURE

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### Responses

• nicla
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8 years ago
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8 years ago
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8 years ago
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4 years ago
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