Introduction to Similar Figure and Pythagorean Theorem Word Problems
Where there is matter, there is geometry.
—JOHANNES KEPLER (1571–1630)
This lesson will review the concepts of similar triangles and the Pythagorean theorem, and will apply these concepts to word-problem solving.
Similar Figures
Similar figures are two or more figures that are the same shape but different sizes. The sides of similar figures are in proportion with each other and the corresponding angles are congruent.
Similar Triangles
There are many types of similar triangles. Some examples are shown next.
To find the measure of missing sides of similar triangles, line up the corresponding parts in a proportion. Then, solve the proportion by cross multiplying. Take the following example.
Example
The triangles in the following figure are similar. What is the value of x in the figure?
Read and understand the question. This question is looking for the value of x in a figure with two similar triangles.
Make a plan. Line up the corresponding sides in a proportion. Then, cross multiply to find the value of x.
Carry out the plan. First, identify the corresponding sides. The side labeled 10 corresponds with the side labeled x, and the side labeled 14 corresponds with the side labeled 28. Set up a proportion using the corresponding parts. Use the proportion
The proportion is
Cross multiply to get 14x = 280. Divide each side of the equation by 14 to get x = 20. The value of x is 20.
Check your answer. To check this solution, substitute x = 20 into the proportion and cross multiply to be sure that the cross products are equal. The proportion becomes 
. Cross multiply to get 280 = 280. The cross products are equal, so this answer is checking.
Other Similar Figures
Similar figures other than triangles can be solved in the same way.
Example
Two quadrilaterals are similar. The shortest side of one quadrilateral is 15 m and the longest is 25 m. If the shortest side of the other quadrilateral is 3 m, what is the measure of the longest side?
Read and understand the question. This question is looking for the value of x in a figure with two similar quadrilaterals.
Make a plan. Line up the corresponding sides in a proportion. Then, cross multiply to find the value of x.
Carry out the plan. First, identify the corresponding sides. The side that is 15 m corresponds with the side that is 3 m, and the side that is 25 m corresponds with the unknown side, or x. Set up a proportion using the corresponding parts. Use the proportion
The proportion is
Cross multiply to get 15x = 75. Divide each side of the equation by 15 to get x = 5. The length of the longest side is 5 m.
Check your answer. To check this solution, substitute x = 5 into the proportion and cross multiply to be sure that the cross products are equal. The proportion becomes 
. Cross multiply to get 75 = 75. The cross products are equal, so this answer is checking.
Tip:Any similar figure problem can be solved using the preceding steps, no matter the number of sides. Simply line up the corresponding parts in a proportion and cross multiply to solve. |
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