4. Understand that a two-dimensional figure is similar to another if the second can be obtained from the first by a sequence of rotations, reflections, translations, and dilations; given two similar two-dimensional figures, describe a sequence that exhibits the similarity between them.
Ever had a favorite piece of clothing made of wool? A cashmere scarf or an angora sock, perhaps? Or maybe you were the victim of your grandmother's itchy holiday sweater, which "accidentally" wound up in the dryer and shrunk to a third its normal size. Your grandma might not have been happy, but it was well worth it. You never got a wool sweater from her again and Mrs. Whiskerson looks more stylish than ever.
Similarity works the same way, only without the dryer sheets. Shapes are called similar when they have the same shape but different sizes. Even though the sweater shrunk, its measurements remained in proportion: if the sleeve was double the neckline, it stayed that way in its miniature form too.
Students should understand that two polygons are similar if they fulfill the following criteria:
- The polygons are the same shape. All corresponding angles in the polygons must be congruent.
- The sizes of the two polygons need not be the same. One polygon could be the size of your palm and the other could be the size of Antarctica.
- The corresponding sides of the polygons must be in proportion to each other. So when comparing corresponding sides, they should all yield the same ratio.
Students should already know that to verify that two polygons are congruent, all we need to do is prove that one can be carried onto the other using translation, reflection, and rotation only. Well, similarity is just the same way with an added bonus of dilation, free of charge.
Dilation is the only transformation that stands in between similarity and congruence because it preserves shape but changes size. Once two similar shapes have been dilated so that they're the same size, they should be congruent and can be moved using the congruence transformations.
In the end, this means that if two shapes can be carried onto one another using nothing but translation, rotation, reflection, and dilation, then they are similar to one another. Make sure your students remember that because it's pretty important. Yes, more important than shrinking more wool sweaters to fit Mrs. Whiskerson. Cats get itchy too, you know.