Question

a rectangle r has vertices p(0,0), q(4,0), r(4,2), and s(0,2). it is transformed into p(1,1), q(5,1), r(5,3), and s(1,3). is the resulting figure similar, and what transformation was performed?

a. yes, the figures are similar; a translation was performed.

b. yes, the figures are similar; a rotation was performed.

c. no, the figures are not similar; a dilation was performed.

d. no, the figures are not similar; a reflection was performed.

Explanation:

Step1: Analyze original rectangle dimensions

Original rectangle \( R \) has vertices \( P(0,0) \), \( Q(4,0) \), \( R(4,2) \), \( S(0,2) \). Length \( PQ = 4 - 0 = 4 \), width \( PS = 2 - 0 = 2 \).

Step2: Analyze transformed rectangle dimensions

Transformed rectangle \( R' \) has vertices \( P'(1,1) \), \( Q'(5,1) \), \( R'(5,3) \), \( S'(1,3) \). Length \( P'Q' = 5 - 1 = 4 \), width \( P'S' = 3 - 1 = 2 \).

Step3: Check similarity and transformation

Since both rectangles have length \( 4 \) and width \( 2 \), their corresponding sides are proportional (ratio \( 1:1 \)), so they are similar. The transformation: each \( x \)-coordinate increases by \( 1 \), each \( y \)-coordinate increases by \( 1 \), which is a translation (shift without rotation/dilation/reflection).

Answer:

a. Yes, the figures are similar; a translation was performed.