Uniformity

If you understand how scales and fractions work, this will be easy. This is a triangle. We copy it like this. We can move the copy and turn it, and shrink it a bit. The right most copy no longer looks exactly like the original, but it still looks alike.

These triangles are similar triangles. This means that all corresponding angles are the same and that the lengths of corresponding sides are proportional. The lengths of all the corresponding sides are in the same ratio. Now, we know the lengths of the two sides of the larger triangle and that of one side of the smaller triangle, and we want to calculate the remaining sides of the smaller triangle. Remember from the definition of similarity, that the lengths of the sides are in the same ratio.

The ratio of 6 to 8 in the larger triangle is the same as the ratio of x to 4 in the smaller triangle. We have the equation. Perhaps you can solve it for x in your head. If not, let's take it step by step. To get x alone on one side, multiply both sides by 4 and simplify on the left side 4 x 6 / 8 is 3.

On the right side, 4 goes away so x = 3. The ratio of 6 to 8 in the larger triangle is the same as the ratio of 3 to 4 in the smaller triangle. You might have all ready noticed that the lengths of the sides of the smaller triangle are exactly half of those in the larger triangle. 4 is half of 8, so x has to be half of 6. If you've done the lesson on scales, you'll notice is the same thing, but we use slightly different symbols and terms.

When two figures are similar, they have the same angles and the lengths of corresponding sides are in the same ratio. Shapes can be moved, rotated, enlarged, or shrunk, or even mirrored. As long as the angles and proportionality don't change, the shapes will be similar.