Question

consider the transformation.
which statement about the transformation is true?
it is isometric because the side lengths remained the same.
it is isometric because all angle measures remained the same.
it is not isometric because the side lengths did not remain the same.
it is not isometric because the angle measures did not remain the same.

Explanation:

Step1: Recall the definition of isometric transformation

An isometric transformation (rigid transformation) preserves side - lengths and angle - measures.

Step2: Check side - lengths

For the trapezoid \(PQRS\): \(PQ = 4\), \(QR=8\), \(RS = 4\), \(SP=12\).
For the trapezoid \(P'Q'R'S'\): \(P'Q'=2\), \(Q'R' = 4\), \(R'S'=2\), \(S'P'=6\).
We can see that \(\frac{PQ}{P'Q'}=\frac{4}{2}=2\), \(\frac{QR}{Q'R'}=\frac{8}{4} = 2\), \(\frac{RS}{R'S'}=\frac{4}{2}=2\), \(\frac{SP}{S'P'}=\frac{12}{6}=2\). The side - lengths are scaled down by a factor of \(2\).

Step3: Check angle - measures

Since the transformation is a similarity transformation (a type of non - isometric transformation when the scale factor is not \(1\)), but angle - measures are preserved in similarity transformations. However, because side - lengths are not preserved (an isometric transformation requires both side - lengths and angle - measures to be preserved), we focus on side - length condition.

Answer:

It is not isometric because the side lengths did not remain the same.