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Similar triangles may or may not have equal side lengths
Triangles ABC and DEF are similar by SSS theorem
<h3>How to determine the similarity statement</h3>The coordinates of the two triangles are:
A (0,0), B (4,0), C (0,2), D (0,0), E (2,0), and F (0,1)
Calculate the lengths of both triangles using the following distance formula
For triangle ABC, we have:
For triangle DEF, we have:
Divide corresponding sides of the triangles to calculate the scale factor k
This gives
Evaluate the quotients
Since the quotients are equal, then the triangles are similar
Hence, triangles ABC and DEF are similar by SSS theorem
Read more about similar triangles at:
A function 
is periodic if there is some constant 
such that 
for all 
in the domain of . Then is the "period" of .
Example:
If
, then we have 
, and so 
is periodic with period 
.
It gets a bit more complicated for a function like yours. We're looking for such that
Expanding on the left, you have
and
It follows that the following must be satisfied:
The first two equations are satisfied whenever
, or more generally, when 
and 
(i.e. any multiple of 4).
The second two are satisfied whenever
, and more generally when 
with (any multiple of 10/7).
It then follows that all four equations will be satisfied whenever the two sets above intersect. This happens when is any common multiple of 4 and 10/7. The least positive one would be 20, which means the period for your function is 20.
Let's verify:
More generally, it can be shown that
is periodic with period
.
Example:
If
It gets a bit more complicated for a function like yours. We're looking for
Expanding on the left, you have
and
It follows that the following must be satisfied:
The first two equations are satisfied whenever
The second two are satisfied whenever
It then follows that all four equations will be satisfied whenever the two sets above intersect. This happens when
Let's verify:
More generally, it can be shown that
is periodic with period