Similar Polygons
Two polygons are similar if these two facts both must be true:
- Corresponding angles are equal.
- The ratios of pairs of corresponding sides must be equal.
(in other words, if they are proportional).
The symbol for "is similar to" is
Similar quadrilaterals
quadrilateral ABCD quadrilateral EFGH
This means:
m
It is possible for a polygon to have one of the above facts true without having the other fact true. The following two examples show how that is possible:
Quadrilaterals that are not similar to one another.
quadrilateral QRST is not similar to quadrilateral WXYZ
Even though the ratios of corresponding sides are equal, corresponding angles are not equal
Quadrilaterals that are not similar to one another.
quadrilateral FGHI is not similar to quadrilateral JKLM
Even though corresponding angles are equal, the ratios of each pair of corresponding sides are not equal
Typically, problems with similar polygons ask for missing sides. To solve for a missing length, find two corresponding sides whose lengths are known. After we do this, we set the ratio equal to the ratio of the missing length and its corresponding side and solve for the variable.
Given that polygon
WXYZ polygon
ABCD, find the missing measure:
The missing measure m is the lenght of . Write a proportion.
|
XY=m, BC=12,
YZ=15, and CD=10 |
| m·10=12·15 |
Find the cross products. |
| 10m=180 |
Multiply |
| m=18 |
Divide each side by 10 |
Find the scale factor from polygon WXYZ to polygon ABCD
| scale factor: |
The scale factor is the constant of proportionality |
A length on polygon WXYZ is times as long as a corresponding length on polygon ABCD.
Let m represent the measure of :
m=18
Congruent polygons have the same shape
and the same size, while similar figures have the same shape buy may have different sizes.
