Greenwood press

120
SERIES

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SIMILARITY
Two figures are similar if they have the same shape, but not necessarily the
same size. More specifically, all of the corresponding sides between two similar
shapes are proportional and all of the corresponding angles are congruent. For
example, most rectangular television screens are similar, since they have a 4-to-3
aspect ratio. That means that conventional television screens are produced so that
the length is 4/3 times the width. The diagonal length of the television screen is
often the reported number in advertisements. Using the 4-to-3 aspect ratio, a tel-
evision screen that has a 25 inch diagonal will have dimensions of 16 inches by 9
inches, and a television screen with a 40 inch diagonal will have dimensions of 32
inches by 24 inches. Notice that the diagonal-to-length ratio is 5 to 4, and the diag-
onal to width ratio is 5 to 3, causing the width, length, and diagonals of every stan-
dard television set to be a multiple of the {3,4,5} Pythagorean triple. 
In 1889, engineers in Thomas Edison’s laboratory established that the 4:3
ratio was the best one for movie screens. It is now being challenged by the 16:9
ratio for high-definition TV sets (HDTV) that use a wider screen than the tradi-
tional one to mimic the wide screens in theaters.
Book covers are examples of two objects that are often not similar. Even
though two books may have rectangular covers with congruent angles, they are
only similar if their side lengths are proportional. For example, a book cover with
dimensions of 6 inches by 3.5 inches is not similar to a book cover with dimen-
sions of 7 inches by 4.5 inches. The corresponding ratios of 7/6 and 4.5/3.5, or
9/7, are not equal.
Similarity is used for many real-world purposes. The film on a movie reel is
projected onto a big screen so that the images appear larger, but in the same pro-
portion. If the screen images were not similar to the slides on the reel, the images
would appear distorted, being either too fat or too long (see Proportions for a
more detailed explanation). An overhead projector serves the same purpose,
allowing images such as a teacher’s handwriting to appear larger on a screen so
that it is easier to read. A telescope and microscope also change the size of
images, making them easier to see while preserving the shape of the original
object. The development of pictures from a camera also uses similarity princi-
ples. As negatives are processed onto photo paper, they expand uniformly in size.
If a picture needs to be enlarged into a poster, then the ratio of the corresponding
sides between the negative and the poster need to be identical. This means that if
the different sizes of photo paper are not similar, then some cropping will occur.