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Scale Drawing Using Proportional Reasoning

Instructor: Laura Pennington

Laura received her Master's degree in Pure Mathematics from Michigan State University. She has 15 years of experience teaching collegiate mathematics at various institutions.

Scale drawings are useful for viewing and analyzing very large or small figures in the world around us. This lesson will explain how to use proportional reasoning to create scale drawings and use those drawings in real-world problem solving.

Proportional Reasoning

Suppose a mother and her daughter get matching purses. The only difference between the purses is that the mother's is larger than the daughter's, and the mother's purse has a strap that the daughter's purse does not. Other than that, they are the exact same shape, color, etc.


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After examining her mother's purse, the daughter decides that she wants a strap on her purse like the one on her mom's purse. They just need to figure out how long the strap should be so that it is proportional to the rest of the purse.

Notice that if we use a ratio, or a comparison of two quantities, to compare the side lengths of the mother's purse to the daughter's purse, they all end up being 3/2.


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Based on this, we can deduce that the ratio of the length of the strap on the mother's purse (4 inches) to the length of the strap that the daughter wants on her purse (x inches), should be equal to 3/2. This gives the following equation:

  • 4/x = 3/2

We can solve this for x to get the strap length.

4/x = 3/2 Multiply both sides by 2 and by x.
8 = 3x Divide both sides by 3.
x = 8/3 This is our answer.

The strap length should be 8/3, or approximately 2.7, inches.

In mathematics, we call this proportional reasoning. Proportional Reasoning involves comparing items using ratios, and using multiplication to identify relationships that we can apply to a given situation. One area we can use proportional reasoning in is scale drawings. Let's take a look!

Scale Drawing Using Proportional Reasoning

In our opening example, we saw that all of the ratios of corresponding sides of the two purses are equal. In mathematics, we say that these two figures are scale drawings of one another, and we call the ratio of their sides the scale factor of the scale drawing.

We can create scale drawings of figures if we have the lengths of the original figure's sides, and a scale factor, a/b, using the following steps:

  1. Multiply each of the side lengths by the scale factor to find the lengths of the corresponding sides that will be on the scale drawing.
  2. Draw a figure that has the same shape as the original figure, but with the side lengths you found in step 1.

For example, consider the triangle shown in the image.


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Suppose we want to create a scale drawing of it using a scale factor of 1/2. No problem! We just take it through our steps!

We first multiply all of the figure's side lengths by 1/2, and then we draw a new figure in the same shape as the original figure with these new side lengths.


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That's not so hard! Let's take a look at another example using proportional reasoning in scale drawings.

Another Example

Suppose that a park ranger is attempting to draw a trail map of the park he works at for some visitors. The park is enclosed by a rectangular trail, and there are also trails along the diagonals of that rectangle. It has a length of 2 miles, a width of 1 mile, and its diagonal trails both have length 2.24 miles. The map is going to be 1/5500 the size of the actual park. Therefore, the map is a scale drawing of the actual park with a scale factor of 1/5500.

Great! We have the shape of the park, the lengths of the trails within the park, and we have the scale factor of 1/5500. That's all the information we need to draw this map!

First, we multiply all of the side lengths and lines in the park by 1/5550, and then we draw those sides and lines in the same shape as the park.


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Perfect! He goes ahead and gives the map to the visitors and sends them on their way.

The visitors decide that they want to hike one loop around the park, and they are wondering how far this will be. Hmmm…assuming they don't know the actual trail lengths of the park, do you have any ideas on how to do find the length of one loop?

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