Question

Help me please:

Prove that the two circles shown below are similar.

Answer 1

Circle X is shown with a center at negative 2, 8 and a radius of 6. Circle Y is shown with a center of 4, 2 and a radius of 3.

Answer 2

ALL circles are similar

you can simply perform a scale and translate transformation to any circle to map it to any other.

The only parameter that defines a circle is the radius

you can scale the radius to make any other circle

Answer 3

how would i scale the radius?

Answer 4

if one circle has raduius 1 and another has radius 2

then scale the first by a factor of 2 - it becomes 'the same' as the other - therefore thay are similar

Answer 5

oh ok

Answer 6

Definition: Two figures are similar if one is the image of the other under a transformation from the plane into itself that multiplies all distances by the same positive scale factor, k. That is to say, one figure is a dilation of the other

Answer 7

yeah the smaller one

Answer 8

If you are expected to do a mathematical proof using transformations, then
The general equation for a circle, centred at (a,b) and radius r is
\(\large (\frac{x-a}{r})^2+(\frac{y-b}{r})^2=1\)
To scale it by a factor of s, we have
\(\large (\frac{x-a}{sr})^2+(\frac{y-b}{sr})^2=1\)
and to further translate it by (h,k), we do
\(\large (\frac{x-a-h}{sr})^2+(\frac{y-b-k}{sr})^2=1\)
What you need to do is to express the circle Y centred at (4,2) radius 3 in standard form.
Then you would scale it with s=2, and translate left by 6, up by 6 (h=-6, k=6) and show that the result, after simplifying the numerator and denominator, is the equation of the large circle.

Answer 9

@mathmate

excellent reply - wish I'd done that myself! :-)

Answer 10

@mrNood
Thank you, we're both trying to help!