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Solving Proportions: More Examples (page 6 of 7)

Sections: Ratios, Proportions, Checking proportionality, Solving proportions


Another category of proportion problem is that of "similar figures".

"Similar" is a geometric term, referring to geometric shapes that are the same, except that one is larger than the other. Think of what happens when you use the "enlarge" or "reduce" setting on a copier, or when you get an eight-by-ten enlargement of a picture you really like, and you'll have the right idea. If you've used a graphics program, think "aspect ratio".

In the context of ratios and proportions, the point is that the corresponding sides of similar figures are proportional.

For instance, look at the similar triangles at the right:

 
The "corresponding sides" are the pairs of sides that "match", except for the enlargement / reduction aspect of their relative sizes. So
A corresponds to a, B corresponds to b, and C corresponds to c.

Since these triangles are similar, the pairs of corresponding sides are proportional. That is, A : a = B : b = C : c. This proportionality of corresponding sides can be used to find the length of a side of a figure.

similar triangles

 

  • In the displayed triangles, the lengths of the sides are given by A = 48 mm, B = 81 mm, C = 68 mm, and a = 21 mm. Find the lengths of sides b and c, rounded to the nearest whole number.

     
    I'll set up my proportions, using ratios in the form (big triangle length) / (little triangle length), and then I'll solve the proportions. Since I have the length of only side
    a for the little triangle, my reference ratio will be A : a.

    First, I'll find the length of b.

      A / a = B / b

      48 / 21 = 81 / b

      b × 48 = 21 × 81 
      48b = 1701

      b = 35.4375

     

    similar triangles

    Now I'll find the length of c. Copyright © Elizabeth Stapel 2001-2011 All Rights Reserved

     

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      A / a = C / c

      48 / 21 = 68 / c

      c × 48 = 21 × 68 
      48c = 1428

      c = 29.75

    For my answer, I could just slap down the two numbers I've found, but those numbers won't make much sense without their units. Also, in re-checking the original exercise, I notice that I'm supposed to round my values to the nearest whole number, so "29.75", with or without units, would be wrong. The right answer is:

      b = 35 mm and c = 30 mm.

  • A picture measuring 3.5" high by 5" wide is to be enlarged so that the width is now 9 inches. How tall will the picture be?

    In other words, the photo lab will be maintaining the aspect ratio; the rectangles representing the outer edges of the pictures will be similar figures. So I set up my proportion and solve:

      (height) : (width): 3.5 / 5 = h / 9

      3.5 / 5 = h / 9

      9 × 3.5 = 5 × h 
      31.5 = 5h

      6.3 = h

    The picture will be 6.3 inches high.

In the first exercise above, the ratios were between corresponding sides, and the proportionality was formed from those pairs of sides; for instance, (length of left-hand slanty side on big triangle) : (length of left-hand slanty side on little triangle) = (length of base on big triangle) : (length of base on little triangle. In the second exercise above, the ratios were between the two different dimensions, and the proportionality was formed from the sets of dimensions: (original height) : (original length) = (enlarged height) : (enlarged length).

For many exercises, you will be able to set up your ratios and proportions in any of various ways. Just make sure that you label things well, clearly define your variables, and set things up in a sensible and consistent manner; this should help you dependably reach the correct solutions. If you're ever not sure of your solution, remember to plug it back into the original exercise, and verify that it works.

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Cite this article as:

Stapel, Elizabeth. "Solving Proportions: More Examples." Purplemath. Available from
    http://www.purplemath.com/modules/ratio6.htm. Accessed
 

 

 

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