Homework Help: Exploring Similar Polygons

Understanding Similar Polygons

A polygon is a 2-dimensional figure with straight sides and angles. In geometry, you'll be asked to compare polygons and determine if they're congruent, similar or neither. Two polygons that are congruent are exactly the same, because the sizes of each of their angles and the lengths of each of their sides are the same. Even if one of the polygons is rotated, translated or reflected so that it's in a different position than the other, they can still be congruent.

In order to be similar, two polygons must have congruent angles and proportional sides. The sides of two polygons are proportional if you can multiply the lengths of all of the sides of one polygon by the same number to get the lengths of all of the sides of the other polygon. For instance, a square with 2-inch sides is similar to a square with 4-inch sides. The sides of the squares are proportional, since each of the sides in the 4-inch by 4-inch square is exactly twice the length of each of the sides in the 2-inch by 2-inch square. The squares also have four congruent angles, since they are all 90 degrees.

Identifying Similarity

Just as congruent shapes may be tricky to identify because one of them is positioned differently from the other, this can also be true with similar shapes. To determine if two shapes are similar, you'll need to consider whether one of the shapes can be rotated (turned), translated (slid) or reflected (flipped) so that it resembles the other. You may need to do more than one of these things to a shape in order to see the resemblance. If you find a resemblance, then you can check to see if the angles of the two polygons are congruent and the lengths of the sides are proportional.

Finding Unknown Lengths

Sometimes, you'll be told that two polygons are similar, and you must use this information to identify the lengths of unknown sides or the sizes of unknown angles. For instance, you might be given two similar polygons, the first of which has four sides with the following lengths:

A = 2 inches

B = 3 inches

C = 4 inches

D = 9 inches

Then, you might be asked to identify the length of Side D in a similar polygon that has three sides with the following lengths:

A = 6 inches

B = 9 inches

C = 12 inches

D = ?

To identify the length of Side D, you would need to observe that Sides A, B and C of the second polygon are each three times the length of the sides in the first polygon. Since the two polygons are similar, then the length of Side D in the second polygon would also be three times the length of Side D in the first polygon. The first polygon's side was nine inches long, and 9 x 3 = 27. As a result, Side D in the second polygon would be 27 inches long.