Question

parallelogram $pqrs$ has coordinates $p(-4, 12), q(4, 12), r(12, - 4),$ and $s(4, - 4)$. parallelogram $pqrs$ has coordinates $p(-12, - 4), q(-12, 4), r(4, 12),$ and $s(4, 4)$. parallelogram $pqrs$ has coordinates $p(-3, - 1), q(-3, 1), r(1, 3)$ and $s(1, 1)$. which transformations describe why parallelograms $pqrs$ and $pqrs$ are similar? parallelogram $pqrs$ was dilated by a scale factor of $\frac{1}{8}$ and then translated 2 units up. parallelogram $pqrs$ was reflected across the $y$-axis and then dilated by a scale factor of $\frac{1}{4}$. parallelogram $pqrs$ was rotated $90^{circ}$ counterclockwise and then dilated by a scale factor of $\frac{1}{4}$. parallelogram $pqrs$ was dilated by a scale factor of $\frac{1}{4}$ and then translated 1 unit right and 2 units down.

Explanation:

Step1: Check dilation

Let's consider the distance between two - points in parallelogram \(PQRS\) and \(P''Q''R''S''\). For example, the length of \(PQ\) with \(P(-4,12)\) and \(Q(4,12)\) is \(d_{PQ}=\vert4 - (-4)\vert=8\). The length of \(P''Q''\) with \(P''(-3,-1)\) and \(Q''(-3,1)\) is \(d_{P''Q''}=\vert1-(-1)\vert = 2\). The ratio of the lengths is \(\frac{d_{P''Q''}}{d_{PQ}}=\frac{2}{8}=\frac{1}{4}\), so there is a dilation by a scale - factor of \(\frac{1}{4}\).

Step2: Check translation

The \(x\) - coordinate of \(P\) is \(-4\) and the \(x\) - coordinate of \(P''\) is \(-3\), so there is a translation of \(1\) unit to the right. The \(y\) - coordinate of \(P\) is \(12\) and the \(y\) - coordinate of \(P''\) is \(-1\), and \(12-(-1)=13\). But if we consider the dilation first, after dilation of \(P(-4,12)\) by a scale factor of \(\frac{1}{4}\), the point becomes \((- 1,3)\). Then, to get to \(P''(-3,-1)\), we need to translate \(1\) unit right (\(-1 + 1=-3\)) and \(2\) units down (\(3-2 = 1\)).

Answer:

Parallelogram \(PQRS\) was dilated by a scale factor of \(\frac{1}{4}\) and then translated \(1\) unit right and \(2\) units down.