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. no, the figures are not similar; a dilation was performed. b. yes, the figures are similar; a translation was performed. c. no, the figures are not similar; a reflection was performed. d. yes, the figures are similar; a rotation was performed.

Explanation:

Step1: Calculate side - lengths of rectangles

For rectangle \(PQRS\) with \(P(0,0)\), \(Q(4,0)\), \(R(4,2)\), \(S(0,2)\):
Length \(PQ=\sqrt{(4 - 0)^2+(0 - 0)^2}=4\), \(QR=\sqrt{(4 - 4)^2+(2 - 0)^2}=2\).
For rectangle \(P'Q'R'S'\) with \(P'(1,1)\), \(Q'(5,1)\), \(R'(5,3)\), \(S'(1,3)\):
Length \(P'Q'=\sqrt{(5 - 1)^2+(1 - 1)^2}=4\), \(Q'R'=\sqrt{(5 - 5)^2+(3 - 1)^2}=2\).

Step2: Check for similarity and transformation type

Since the corresponding side - lengths are equal (\(PQ = P'Q'\) and \(QR=Q'R'\)), the rectangles are congruent (a special case of similar figures with a scale factor of 1).
To go from \(PQRS\) to \(P'Q'R'S'\), we can observe that \(x\) - coordinates of all points of \(PQRS\) are increased by 1 and \(y\) - coordinates of all points are increased by 1. This is a translation.

Answer:

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