Question

Which method is valid for proving that two circles are similar?

Answer 1

First I'm going to go through the choices with you and evaluate
each one.  Then after that, I'm going to hand you a secret that
I promise is going to knock your socks off.


a- Calculate the ratio of the diameter to the radius for each circle
and show that they are equal.
-- That won't tell you anything. The ratio of the diameter
to the radius of EVERY circle is 2 .

b- Calculate the ratio of degrees to the circumference for each circle
and show that they are equal.
-- That doesn't tell you anything.  The circumference
of EVERY circle subtends a central angle of 360°.

c- Calculate the ratio of the área to the circumference for each circle
and show that they are equal.
-- That doesn't tell you anything.  The ratio of the area
to the circumference of EVERY circle is (radius/2). 
They're only equal if the circles are the same size.

d- Calculate the ratio of the diameter to the circumference for each circle
and show that they are equal.
-- That doesn't tell you anything.  The ratio of the diameter
to the circumference of EVERY circle is  1/pi.   If the ratio isn't
1/pi, then you're not looking at a circle. 

None of these choices tells you whether the two circles are similar.
What are you going to do ?  How can you tell ? ?

Here's the surprise I promised you.
Beware of flying socks:

All circles are similar to all other circles.

Good night.