Two polygons whose corresponding angles are congruent and corresponding sides are in equal proportion are called similar polygon. The symbol to denote that two polygons are similar is ~.
Similar polygons have same shape but are different in size. For example as shown in the below figure rectangle ABCD ~ rectangle PQRS. In these two rectangles:
?A is congruent to ?P
?B is congruent to ?Q
?C is congruent to ?R
?D is congruent to ?S
Properties of similar polygons:
(refer the above figure)
1) Corresponding angles are same: so in the above figure
?A=?P, ?B=?Q, ?C=?R, ?D=?S
Corresponding exterior angles are also same.
2) Corresponding sides ratio are same: each pair of corresponding sides are in same proportion.
3) Corresponding diagonals are in the same proportion
4) Ratio of area of two similar polygons is the square of the ratio of sides. As the sides are in ratio of 2:1 therefore area is in the ratio of 4:1.
Solved Problems on Similar Polygon:
Example 1: Find the value of x, y, and the measure of angle B.
Solution: To find the value of x and y, write proportions involving corresponding sides. Then use cross products to solve.
6x = 24 2y = 30
x = 4 y = 15
To find angle B, note that angle B and angle Q are corresponding angles.
By definition of similar polygon
angle B = angle Q = 77o.
Example 2:
In ?ABC it is given that DE||BC and AD=2, AB=6, AC=9 then find AE.
Solution: ?ADE~?ABC because DE||BC
Therefore
?B=?D
?C=?E
And ?A is common so from the these we can say that ?ADE~?ABC
So from the similar polygon rule
6AE=18
AE=3
Related Tags
Study of Similar Polygons, Examples on Similar Polygons, Solved problems on Similar Polygons