Condition For Similarity
Statement: Polygons A 1 A 2 A 3 … A n and B 1 B 2 B 3 … B n are similar polygons.
Condition: The ratio of the corresponding sides of the polygons is constant, i.e.
B 1 B 2 A 1 A 2 = B 2 B 3 A 2 A 3 = ⋯ = B n − 1 B n A n − 1 A n = B n B 1 A n A 1
For the statement to be true, the given condition is ___________.
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1 solution
Can you add an explicit example where the condition is not sufficient?
For example, if we were dealing only with triangles, then the condition is necessary and sufficient.
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Thanks, I have updated the solution with examples for both conditions.
For two polygons to be similar ,
Both the conditions must be satisfied for the polygons to be similar. If either of the conditions is not satisfied, then the polygons do not have to be similar.
For example, consider a square and a rhombus; or a regular decagon and 5-pointed star. They satisfy condition 1, but not condition 2.
Note that any amount of uniform scaling, translation, reflection or rotation cannot transform the square into the rhombus, therefore the square and the rhombus are not similar. Similarly the decagon and the star are not similar either.
We see the first condition in itself is necessary but not sufficient for the polygons to be similar. □
The second condition is not sufficient on its own either. We can have a square and a rectangle which satisfy condition 2 but not condition 1.
We see that these two figures are not similar.