Given: Circle [tex][tex]$X$[/tex][/tex] with radius [tex][tex]$r$[/tex][/tex] and circle [tex][tex]$Y$[/tex][/tex] with radius [tex][tex]$s$[/tex][/tex]

Prove: Circle [tex][tex]$X$[/tex][/tex] is similar to circle [tex][tex]$Y$[/tex][/tex].

[tex]\[
\begin{array}{|c|c|}
\hline
\text{Statements} & \text{Reasons} \\
\hline
1. \text{Circle with center } X \text{ has radius } r & \text{Given} \\
\hline
2. \text{Circle with center } Y \text{ has radius } s & \text{Given} \\
\hline
3. \text{Translation is a similarity transformation} & \text{Definition of similarity transformation} \\
\hline
4. \text{Dilation is a similarity transformation} & \text{Definition of similarity transformation} \\
\hline
5. \text{A composition of similarity transformations maps circle } X \text{ to circle } Y & \text{Composition of similarity transformations} \\
\hline
\end{array}
\][/tex]



Answer :

To prove that Circle [tex]\( X \)[/tex] with radius [tex]\( r \)[/tex] is similar to Circle [tex]\( Y \)[/tex] with radius [tex]\( s \)[/tex], we need to show that there exists a similarity transformation that maps Circle [tex]\( X \)[/tex] to Circle [tex]\( Y \)[/tex]. Here is a detailed step-by-step proof using the given information and logical reasoning:

### Proof:

1. Statement: Circle [tex]\( X \)[/tex] has a radius [tex]\( r \)[/tex].
Reason: Given

2. Statement: Circle [tex]\( Y \)[/tex] has a radius [tex]\( s \)[/tex].
Reason: Given

3. Statement: Translation is a similarity transformation.
Reason: By definition, translation moves every point of a figure or a space by the same distance in a given direction without changing its shape or size.

4. Statement: Dilation is a similarity transformation.
Reason: By definition, dilation multiplies distances from a fixed center (the center of dilation) by a common scale factor, resulting in a shape that is similar to the original.

5. Statement: Translation followed by dilation maps Circle [tex]\( X \)[/tex] to Circle [tex]\( Y \)[/tex].
Reason: A composition of similarity transformations is also a similarity transformation. First, translate Circle [tex]\( X \)[/tex] so that its center coincides with the center of Circle [tex]\( Y \)[/tex]. Next, apply a dilation centered at this common center with a scale factor of [tex]\( \frac{s}{r} \)[/tex].

6. Statement: Circle [tex]\( X \)[/tex] is similar to Circle [tex]\( Y \)[/tex].
Reason: Since we can map Circle [tex]\( X \)[/tex] to Circle [tex]\( Y \)[/tex] using a composition of similarity transformations (translation and dilation), the circles are similar by definition of similarity transformations.

### Conclusion:
Since we have shown that a series of similarity transformations (translation followed by dilation) can map Circle [tex]\( X \)[/tex] to Circle [tex]\( Y \)[/tex], we conclude that Circle [tex]\( X \)[/tex] is similar to Circle [tex]\( Y \)[/tex].