Question

for the graph to the right, describe the composition of transformations that maps \\( \triangle pby \\) to \\( \triangle ngz \\).
which composition of transformations below maps \\( \triangle pby \\) to \\( \triangle ngz \\)? select the correct choice below
and fill in the answer boxes to complete your choice.
(type an integer or a decimal.)
\\( \bigcirc \\) a. \\( d_\square \circ r_{y - \text{axis}} \circ r_{(180^{\circ}, o)}(\triangle pby) \\)
\\( \bigcirc \\) b. \\( d_\square \circ r_{(270^{\circ}, o)}(\triangle pby) \\)
\\( \bigcirc \\) c. \\( d_\square \circ r_{(180^{\circ}, o)}(\triangle pby) \\)
\\( \bigcirc \\) d. \\( d_\square \circ r_{y = -x}(\triangle pby) \\)

Explanation:

Step1: Analyze the transformation sequence

First, consider the rotation. If we rotate \(\triangle PBY\) \(270^{\circ}\) clockwise about the origin \(r_{(270^{\circ},O)}\), the orientation of the triangle changes.

Step2: Analyze the dilation

After rotation, we need to check the side - length ratios. Let's assume the side - lengths of \(\triangle PBY\) and \(\triangle NGZ\). If we assume the side - length of a side of \(\triangle PBY\) is \(2\) units and the corresponding side of \(\triangle NGZ\) is \(1\) unit. The dilation factor \(k\) is given by the formula \(k=\frac{\text{length of side in image}}{\text{length of side in pre - image}}\).

Let's assume a side of \(\triangle PBY\) (e.g., \(PB\)) has a length. Counting the grid units, if \(PB = 4\) units (from \((- 2,3)\) to \((-2,-1)\)) and the corresponding side \(NG\) (from \((2,1)\) to \((2,-1)\)) has a length of \(2\) units. The dilation factor \(D_{\frac{1}{2}}\) (since \(\frac{2}{4}=\frac{1}{2}\))

Answer:

B. \(D_{\frac{1}{2}}\circ r_{(270^{\circ},O)}(\triangle PBY)\)