Photocopier
Have you ever noticed that when you place a picture on a photocopier turned one way the copy comes out flipped upside down? Have you ever tried to enlarge a copy of an object to fit into an special frame, and had to enlarge it several times before you got the size "just right?" Have you ever made a photocopy of a book cover or of your hands? Well, if you have witnessed or experienced any of these things, then you have been involved in some of the everyday mathematics that surrounds us.
Congruency, Similarity, and Nonsimilarity
The photocopier can produce copies that have virtually the same size as the original item copied. The two objects—original and its copy—are said to be mathematically congruent to each other. Two 2-dimensional objects are congruent if the objects have the same shape and the same size. On the photocopier, the copy comes out as a reflection of the original (flipped upside down), so you may have to flip the copy to fit it on top of the original to see the perfect match in shape and size.
The photocopier also can produce copies of objects that are smaller (reduction) or larger (magnification) than the original object. Mathematically, we refer to the process of making this different-sized copy as a dilation. The copy has the same shape as the original, but not the same size. Certain corresponding measurements (such as the left side on an original and the left side on a copy) are multiples or fractions of the original. Objects that fit this description are referred to as mathematically similar shapes. The word "similar" is being used here in a more restricted way than you probably use it in everyday conversations.
In the illustration below are four pairs of similar shapes. For example, A′ is the copy of A. Do these shapes—A and A′, B and B′, C and C′, and D and D′—seem similar?
In contrast, the illustration at the top of the next page shows two pairs of shapes that are almost but not quite similar. Shape E is almost similar to E′ and F is almost similar to F′. Yet the ratios of width to height are different in the two pairs, and cause the shapes to fail the test of being similar.
Examples of Similarity
A few examples of similarity on a photocopier illustrate the mathematics behind reductions and enlargements.
Copy of a Triangle. Imagine your original shape is a triangle with base length that is 2 inches and height that is 1 inch. (See the single small triangle on the left in the table.) If you enlarge the triangle with a photocopier and measure the base length and height of the copy, you can compare those new lengths to the original values. You will find that you can either multiply or divide by one specific number to get the measurement of the copy based upon the measurement of the original.
In this example, the length of the base of the copy of the triangle is 4 inches, yet the original base length was 2 inches. What number might you multiply times 2 to get 4? So, if you multiply the same number, 2, times the height measure of the original triangle (1 inch), you will get 2 inches (2 × 1 inch), the height of the similar copy. Since multiplying the measurements of the original triangle by 2 will yield the measurements of the new triangle, we say we have used a scale factor of 2.
There are other ways of referring to this same scale factor of 2. We might say that the ratio of side lengths of the similar copy to the side lengths of the original is 2:1, or 2-to-1. When we use a photocopier, we use percentages to refer to the scale factor. In this example, the scale factor on the photocopier would be represented as 200%:100%, or we would simply choose the enlargement (or magnification) factor to be 200%.
| COMPARISON OF AN ORIGINAL AND ITS COPY |
| Original Triangle | Formula | Copy of Triangle
(scale factor of 2) |
| 2 inches | Base length (b) | 4 inches |
| 1 inch | Height (h) | 2 inches |
| 0.5(2)(1) = 1 sq. in. | Area = 0.5bh | 0.5(4)(2) = 4 sq. in. |
 | |  |
Determining the relationship between the area of the original shape and its similar copy is more challenging. You will need to give thought to the overall size of the photocopy, because you should make sure that the size of paper in the photocopier is large enough to hold the copy of the original shape. So, if you have doubled the base length and doubled the height of the original triangle, do you think the area will also be doubled?
There are at least two ways to find the answer, as shown in the table. Calculate the area with a formula or discover the area by tessellating copies of the smaller triangle until you cover the similar copy of the original triangle. ("Tessellating" means to assemble the smaller triangles so they adjoin one another with no gaps in between.)
Reducing a Photograph. Suppose one of your friends has given you a 5- by-7-inch photograph. Although you plan to place the photo into a frame, you would like to have a wallet-sized copy of this photo, too. The photo is not copyrighted to prohibit copying, so you find a color photocopier to make your copy. Your wallet will hold a 2.5-by-3.5-inch photo. Which setting will you use on the photocopier to make your copy?
There are at least two ways to find the answer without guessing at the reduction. If the original has a width of 5 inches, and your copy requires a width of 2.5", the ratio is 5:2.5 which equals 2:1. Since you are making a reduction, you must reverse the order to 1:2. Hence, you will need to use 50% as the reduction factor on the photocopier.
The 50% setting works for the width of the photo, but will it work for the length of the photo as well? If we want a similar copy, we realize that the scale factor for the length of the photo must be the same as the scale factor of the width. Thus, the photocopier setting should be correct at 50%. Will a 50% reduction from 7 inches in length yield a new length of 3.5 inches? Yes, 50% of 7 is 3.5.
Enlarging a Photograph. Now suppose you decide to also enlarge the 5- by-7-inch photo to make it fit into an 8-by-10-inch frame. Can you figure out the setting for the photocopier?
First, try changing the 5-inch width to 8 inches. So the scale factor is eight-fifths, or 1.6. On the photocopier, you would choose 160%. This same scale factor should give the desired length of 10 inches on the photocopy.
Will a magnification of 160% from 7 inches in length yield a new length of 10 inches? Not quite, because 160% of 7 is 11.2 inches. That poses a problem if you are using 8.5-by-11-inch paper in the photocopier—your copy will be 0.2 inches too long! So what do you do? You could switch to a larger paper tray in the photocopier. However, your copy will be too large to fit into the 8-by-10 photo frame. You can also make a decision to cut off part of the length of the photo, or use a smaller width and try for a photo-copy that will be similar to the original. If you choose the latter option, you will decide on a slightly smaller magnification setting on the photocopier.
Remember there is more than one way to solve a problem, so the drawing below shows another approach you may try when you make measurements to determine a proper setting for a reduction or magnification on the photocopier.
Congruency, Equality, and Similarity; Percent; Scale Drawings and Models; Ratio, Rate, and Proportion.
Bibliography
Bloomfield, Louis A. How Things Work: The Physics of Everyday Life. New York: John Wiley & Sons, Inc., 1997.
Garfunkel, Solomon, Godbold, Landy, and Pollock, Henry. Mathematics: Modeling our world. ARISE Course 3. Cincinnati: South-western Educational Publishing, 1999.
O'Daffer, Phares G., and Clemens, Stanley R. Geometry: An Investigative Approach, 2nd ed. Reading, MA: Addison-Wesley, 1992.
Serra, Michael. Discovering Geometry: An Inductive Approach. Berkeley: Key Curriculum Press, 1997.
Walton, Stewart, & Walton, Sally. Creative Photocopying: Using the Photocopier for Crafts, Design, and Interior Decorations. New York: Watson-Guptill Publications, 1997.
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