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# 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 objectsoriginal and its copyare 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 shapesA and A, B and B, C and C, and D and Dseem 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-to1. 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 photocopieryour 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.

see also Congruency, Equality, and Similarity; Percent; Scale Drawings and Models; Ratio, Rate, and Proportion.

Iris DeLoach Johnson

## 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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# Photocopying

Photocopying is the process of photographically reproducing a document of text, illustrations, or other graphic matter. The most common photocopying method used today is called xerography (from the Greek words for "dry" and "writing").

## The process of photocopying

The mechanics of photocopying is based on the principle of photoconductivity (when certain substances allow an electric current to flow through them when light is applied). For example, when light is absorbed by some of the electrons (particles that have a negative charge) that make up selenium (a nonmetallic chemical element that is used in the photocopying process), the electrons are able to pass from one atom to another when voltage is applied. When the light source is taken away, the electrons lose their mobility or ability to move.

## Words to Know

Electrostatics: Relating to painting with a spray that utilizes electrically charged particles to ensure a complete coating.

Toner: A material that carries an electrical charge opposite to that of a photoconducting surface that is added to that surface in a copy machine.

Xerography: A method of copying that uses dry powder, electric charge, and light to fuse an image onto paper.

During the process of photocopying, the round drum (usually made of aluminum) inside the copier is coated with a layer of selenium that is given a positive electrical charge. After placing the document to be copied on the glass-topped surface of the copier, a light exposes the image of the

document onto the drum. This causes the positive charge on the selenium-coated drum to fade except from the area to be copied. This area remains charged.

The negatively charged toner (ink) is then sprayed onto the drum, which forms an exact duplicate copy of the document. After that process is completed, a sheet of copy paper is passed by the drum at the same time that a positive electric charge is passed under the paper. The positive charge attracts the negatively charged toner image on the drum and the toner sticks in the same pattern onto the paper. Heat is quickly applied

to the copied image on the paper and it adheres the toner permanently to the paper.

Over the years, many improvements in the photocopying machine have taken place. Some enhancements include sorting and collating (arranging in order), enlarging or reducing the copied material, printing on both sides of the paper, and reproducing in color.

## Inventor of photocopying

Xerography was invented by American physicist Chester F. Carlson (19061968) in 1938. After earning his physics degree from the California Institute of Technology in 1930, Carlson accepted a job working for the P.R. Mallory Company, an electronics business in New York. Working in the patent department, Carlson was frustrated by the difficulty of obtaining copies of patent drawings and specifications. He decided to use his time away from work to find a solution to the problem.

Focusing on the concept of electrostatics, Carlson spent four years before succeeding in production his first "dry-copy." The first successful copy was a notation of the date and location that read "10.22.38 Astoria." (Carlson lived in Astoria, Queens, New York at the time.) In 1940, Carlson obtained the first of many patents for his xerographic process. Wanting to find a company that would help him develop and market his idea, Carlson began showing his solution to many organizations. After more than twenty firms turned down his invention, Carlson finally reached an agreement in 1944 with the Battelle Memorial Institute, a nonprofit research organization. Three years later, the Haloid Company (later the Xerox Corporation) became a partner in the development of the xerography technology. Finally, after years of development, the first office copierthe Xerox 914was introduced in 1959.

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## photocopying

photocopying, process whereby written or printed matter is directly copied by photographic techniques. Generally, photocopying is practical when just a few copies of an original are needed. When many copies are required, printing processes are more economical. However, when a printing process is used, the master or stencil required can sometimes be produced by photocopying. Principal photocopying processes include silver halide, transfer, plan, thermographic (see thermography), and electrostatic (e.g., xerography, which has become so widespread that the process is popularly almost synonymous with photocopying). Two well-known silver halide processes, photostating (see photostat) and microfilming, use cameras to make photographic copies of an original. Microfilming generates copies that are from 1/12 to as little as 1/100 the size of the originals, allowing great economy in space and materials when long-term storage is necessary. Microfilms are read by either projecting them or photographically printing them as enlargements. In transfer processes the original is placed in contact with negative paper and exposed to light. In the diffusion transfer the negative is developed while in contact with the positive. During development the chemicals forming the image in the negative diffuse to the positive, producing an image there. In gelatin transfer the negative is developed and then pressed against positive paper. A dyed gelatin on the negative is picked up by the positive, producing an image on it. Transfer methods are less expensive than silver halide processes, but the image produced by the former deteriorates with time. Plan copying is used to copy materials such as architects' drawings and engineers' plans. In one variety of plan copying known as the whiteprint process, an original is made on translucent paper. The paper is placed over a sheet coated with a diazo compound and exposed to a source of ultraviolet light. The compound covering the area that is exposed to the light decomposes. The compound shielded from the ultraviolet rays by the dark areas of the original can be developed to form a positive image. The blueprint process, another method of plan copying, has been largely superseded by whiteprints, which are of better quality and cost approximately the same.

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## photocopying

photocopying Reproduction of words, drawings, or photographs by machine. In a photocopying machine, a light shines on the item to be copied, and an optical system forms an image of it. Various techniques may be used to reproduce this image on paper. In a modern plain-paper copier, the image is projected onto an electrically charged drum, coated with the light-sensitive element selenium. Light makes the selenium conduct electricity, so bright areas of the drum lose their charge. The dark areas, which usually correspond to image detail, retain their charge, and this attracts particles of a fine powder called toner. Electrically charged paper in contact with the drum picks up the pattern of toner powder. A heated roller fuses the powder so that it sticks to the paper and forms a permanent image.

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## photocopy

pho·to·cop·y / ˈfōtəˌkäpē/ • n. (pl. -cop·ies) a photographic copy of printed or written material produced by a process involving the action of light on a specially prepared surface. • v. (-cop·ies, -cop·ied) [tr.] make a photocopy of. DERIVATIVES: pho·to·cop·i·a·ble / -ˌkäpēəbəl/ adj.

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## photocopier

pho·to·cop·i·er / ˈfōtəˌkäpēər/ • n. a machine for making photocopies.

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## photocopier

photocopier •Dampier •Napier, rapier, tapir •Shakespeare • sepia • Olympia •copier • compeer • photocopier •cornucopia, dystopia, Ethiopia, myopia, subtopia, Utopia

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## photocopy

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