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This essential trains for: SAT-I, GMAT, AMC-8, SAT-II, Math Kangaroo 5-6, Math Kangaroo 7-8.

Similar figures have side lengths that are directly proportional.

Think of similar figures as enlargements/reductions to scale of images.

The side lengths that are proportional correspond to identical neighborhoods of the image.

In two similar triangles:

figure

The proportionality of the sides is written as:

equation

It is not only the sides that are proportional but all the segments that are somehow related to the triangle: medians, altitudes and any other cevians.

Since two corresponding altitudes of the same triangles satisfy:

equation

the areas of the similar triangles are in a ratio:

equation

The ratio of any two corresponding lengths is called the similarity ratio between the two figures.

If two figures are similar in a ratio of r then their areas are in a ratio of r2.

If two solids are similar in a ratio of r then their surface areas are in a ratio of r2 and their volumes are in a ratio of r3.

Example:

A cube has a side length equal to 1 unit. If the side length is tripled, what is the surface area/volume of the resulting larger cube?

Since the side lengths are in a ratio of 1:3, the surface areas are in a ratio of 1:9 and the volumes are in a ratio of 1:27.

A cube has 6 faces. If each has area 1 square unit, the total surface area of the small cube is equal to 6 square units.

The surface area of the larger cube is nine times as large:

equation

The volume of the little cube is equal to 1 cubic unit.

The volume of the larger cube is twenty-seven times as large:

equation