91Ó°ÊÓ

Prove the Reflections in Parallel Lines Theorem (Theorem 4.2). Given \(\quad\) A reflection in line\(^\ell\) \(\operatorname{maps} \overline{J K}\) to \(\overline{J^{\prime} K^{\prime}},\) a reflection in line m maps \(\overline{\mathrm{J}^{\prime} \mathrm{K}^{\prime}}\) to \(\overline{\mathrm{J}^{\prime \prime} \mathrm{K}^{\prime \prime}}\) and\(^\ell\) m. Prove \(\quad\) a. \(\overline{\mathrm{KK}^{\prime \prime}}\) is perpendicular to \(\ell\) and \(\mathrm{m}\) \(\qquad\) \(\quad\) b. \(\mathrm{KK}^{\prime \prime}=2 \mathrm{d},\) where d is the distance between \(\ell\) and \(\mathrm{m} .\)

\(\overline{\mathrm{PQ}},\) with endpoints \(\mathrm{P}(1,3)\) and \(\mathrm{Q}(3,2),\) is reflected in the \(y\)-axis. The image \(\overline{\mathrm{P}^{\prime} Q^{\prime}}\) is then reflected in the \(x\)-axis to produce the image \(\overline{\mathrm{P}^{\prime \prime} \mathrm{Q}^{\prime \prime}}.\) One classmate says that \(\overline{\mathrm{PQ}}\) is mapped to \(\overline{\mathrm{P}^{\prime \prime} \mathrm{Q}^{\prime \prime}}\) by the translation \((\mathrm{x}, \mathrm{y}) \rightarrow(\mathrm{x}-4, \mathrm{y}-5) .\) Another classmate says that \(\overline{\mathrm{PQ}}\) is mapped to \(\overline{\mathrm{P}^{\prime \prime} \mathrm{Q}^{\prime \prime}}\) by a \((2 \cdot 90)^{\circ},\) or \(180^{\circ}\), rotation about the origin. Which classmate is correct? Explain your reasoning.

ANALYZING RELATIONSHIPS it possible for a figure to have 180 rotational symmetry but not \(90^{\circ}\) rotational symmetry? Explain your reasoning.

THOUGHT PROVOKING an rotations of \(90^{\circ}, 180^{\circ}\) , \(270^{\circ},\) and \(360^{\circ}\) be written as the composition of two reflections? Justify your answer.

Does the order of reflections for a composition of two reflections in parallel lines matter? For example, is reflecting \(\Delta \mathrm{XYZ}\) \(\ell\) and then its image in line m the same as re ecting \(\Delta \mathrm{XYZ}\) in line m and then its image in line \(\ell\) ?

REASONING Use the coordinate rules for counterclockwise rotations about the origin to write coordinate rules for clockwise rotations of \(90^{\circ}, 180^{\circ}\) , or \(270^{\circ}\) about the origin.

WRITING Explain how to use translations to draw a rectangular prism.

MATHEMATICAL CONNECTIONS Re?ect ?MNQ in the line y=?2x

MAKING AN ARGUMENT A translation maps GH to G'H'. Your friend claims that if you draw segments connecting G to G' and H to H', then the resulting quadrilateral is a parallelogram. Is your friend correct? Explain your reasoning.

USING STRUCTURE A polar coordinate system locates a point in a plane by its distance from the origin O and by the measure of an angle with its vertex at the origin. For example, the point \(\mathrm{A}\left(2,30^{\circ}\right)\) is 2 units from the origin and \(\mathrm{m} \angle \mathrm{XOA}=30^{\circ}\) . What are the polar coordinates of the image of point A after a \(90^{\circ}\) rotation? a \(180^{\circ}\) rotation? a \(270^{\circ}\) rotation? Explain.