Question

George is given two circles, circle 0 and circle X. If he wants to prove that the two circles are similar what would be the correct first step in his proof

Answer 1

Explanation:

Similarity is a shape preserving transformation. This may include dilation, or shift horizontal or shift vertical or rotation through a point.

Consider the two circles O and X given

Let radius of O be r, and radius of X be r1.

Let us assume that r = kr1

Then k is a real rational value, and always positive.

Thus on comparison we find that since radius of circle O is r, circle X is obtained form Circle O as a transformation by dilating the circle with a factor of k.

Since dilation results in a similar figure, we have circles O and X are similar.

Answer 2

Hello!

Two circles are always similar, in fact, all the circles will be always similar since the geometric form of all circles involves infinite angles and sides.

Two circles will be always similar because of the ratio proportionality.

Ratio proportionality can be proved by making it concentric. You will be always able to find a number that multiplies the smaller circle ratio to make it exactly equal to the bigger one, so, doing that, all we can safely assume that all the circles are similar (and proportional).

By using the circumference definition, we have:

`C_(0) = 2 * \pi * r_(0) \\C_(X) = 2 * \pi * r_(X) \\\\\\C_(X) = 2 * \pi * r_(X) = C_(0) = 2 * \pi * r_(0)\\`

So,

`(r_(X))/(r_(0))`

Have a nice day!