Lesson 1: Similar Polygons - Determining Similarity with Side Lengths

   Constructing Knowledge

For two polygons to be similar, their corresponding side lengths must be proportional throughout the entire figure. The polygons below, from the previous page, have corresponding angle measures that are congruent. To determine if they are mathematically similar, corresponding side length ratios must be compared.


For the polygons above to be similar:

\(\frac{\text{PQ}}{\text{AB}}=\frac{\text{QR}}{\text{BC}}=\frac{\text{RS}}{\text{CD}}=\frac{\text{PS}}{\text{AD}}\)

If the ratios are all equal and the corresponding angle measures are congruent, the polygons are similar.

When determining polygon similarity:

  • all corresponding side length ratios must be compared. If even one ratio is different, the polygons are not similar.
  • If the polygons are not shown with the same orientation, redrawing one of the figures may help you identify corresponding side lengths and angle measures.

   Multimedia

A video describing similarity between regular polygons is provided.



EXAMPLE 1

Are the polygons below similar?


Solution


Step 1: Identify all corresponding sides.

relative position Polygon ABCD Corresponding side on
Polygon PQRS
AB PQ
BC QR
CD RS
AD PS

Step 2: Calculate each of the corresponding side ratios and compare.

relative position Polygon PQRS:
Polygon ABCD 		Corresponding side ratio
\(\frac{\text{PQ}}{\text{AB}}=\frac{3.15}{2.1}\) = 1.5
\(\frac{\text{QR}}{\text{BC}}=\frac{1.65}{1.1}\) = 1.5
\(\frac{\text{RS}}{\text{CD}}=\frac{1.5}{1}\) = 1.5
\(\frac{\text{PS}}{\text{AD}}=\frac{3.3}{2.2}\) = 1.5

Because the ratios are the same for all corresponding side lengths, the polygons are similar.


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