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Ken Ward's Mathematics Pages

Geometry

Proportion and Similar Figures, Areas, Etc

Any shape is similar to another if all corresponding measurements are in proportion to a constant, called a scaling factor. Similar figures are such that by zooming in or zooming out of one figure, we obtain a figure identical to the other figure. Similar figures have the same shape.

Similar Triangles

Triangles are similar when their angles are the same. (We don't need to say they are corresponding, because if two angles are the same, the third one will also be the same, and the sides will be corresponding).  
In the diagram below:
proportionTriangles.gif
ABC is a triangle, and DE is parallel to AB. DEC is therefore similar to ABC. The height of DEC is h1, and the height of ABC is h. Because they are similar triangles, their corresponding dimensions are proportional to a scaling factor, so:
h1Divh=DEdivAB.gif
So:

h1=DEh1DivAb.gif
For instance, if h=4, h1=2, AB=4 then h1=2/4*2=2, that is it is half the size. Or because h1/h=2/4, the scaling factor is 1/2.

However, the area of the big triangle is 4*4//2=8, and the area of the smaller triangle is 2*2/2=2. That is, the ratio of the areas is 1:4.
In general, if s is the scaling factor, and DE=b1 and AB=b, then
bDivh-b1Divh1=s.gif
If A1 is the area of DEC, and A is the area of ABC, then:
proportionA1DivA.gif
Now, because h1/h=b1/b=s, then
h1DivXb1Divb..bmp

Any Similar Figures

The ratio of the areas of any similar figures are in proportion to the scaling factor squared.

Suppose that the two patterns are similar; that is, all their corresponding dimensions are proportional to the scaling factor, s. Suppose we represent the areas by small squares (pixels) of side x, so the area of each square is x2, then the area, A1, covered by one square in one of the similar figures will be covered by an area, A2, sx*sx in the other figure. We can, of course, make our little squares so small, they exactly represent the areas. The ratio of the areas, A1 and A2 is therefore:
A1DivA2=xPow2.gif
And any similar figures, or patterns are therefore proportional to the scaling factor squared.








Ken Ward's Mathematics Pages


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