Exercise 2 Page 588

Let's consider what information is given and what we want to prove. Given:& Circle C with center (2,1) and radius 1 & Circle D with center (0,3) and radius 4 Prove:& Circle C is similar to Circle D.

We have also been given an incomplete paragraph proof which we need to complete. Let's have a look at the first sentence of the proof. Map CircleC to CircleC' by using the (x,y)→ so that Circle C' and CircleD have the same center at( , ). Let's use a graph to help us. In the graph Circle C' has the same radius as Circle C and the same center as Circle D.

We can map Circle C to Circle C' by translating it.

This is a translation of 2 units left and 2 units up. (x,y)→ (x -2,y +2) We know that Circle D has its center at (0,3). That is also the center of Circle C'. We can now complete the first sentence of the proof. Map CircleC to CircleC' by using the translation (x,y)→ (x- 2,y+2) so that Circle C' and CircleD have the same center at( ,3). Let's now have a look at the second sentence of the paragraph proof. Dilate Circle C' using a center of dilation ( , ) and a scale factor of . After we have dilated Circle C' we want it to coincide with Circle D. Since both circles have the same center, (0,3), this must also be the center of dilation.

With a scale factor of 4 the circles coincide. Let's complete the second sentence in the proof. Dilate Circle C' using a center of dilation ( ,3) and a scale factor of 4. We are ready to take a look at the final sentence of the paragraph proof. Because there is a transformation that maps Circle C to Circle D, Circle C is Circle D. We mapped Circle C to Circle D by a translation followed by a dilation. These are similarity transformations. Two figures that can be mapped onto each other using only similarity transformations are said to be similar. Therefore, Circle C is similar to Circle D. Let's complete the last sentence. Because there is a similarity transformation that maps Circle C to Circle D, Circle C is similar Circle D.