Ticutormetians Writing a ruld to deacribe a retalion Triangle RST is rotated \( 90^{\circ} \) counterclockwise about the origin. The result is \( \Delta R^{\prime} S^{\prime} T^{\prime} \), as shown below. (a) The arrows below show that the coordinates on the left are mapped to the coordinates on the right. Fill in the blanks to give the coordinates after the rctation. original coordinates - final coordinates \[ \begin{aligned} R(-8,-2) & \rightarrow R^{\prime}(\square) \\ S(1,-4) & \rightarrow S^{\prime}(\square) \\ T(-2,-6) & \rightarrow r^{\prime}(\square) \end{aligned} \] (b) Choose the general rule below tha: describes the rotation mapping \( \triangle R S T \) to \( \triangle R^{\prime} S^{\prime} T^{\prime} \). \( (x, y) \rightarrow(-y, x) \) \( (x, y) \rightarrow(-x,-y) \) \( (x, y) \rightarrow(y,-x) \) \( (x, y) \rightarrow(x,-y) \) \( (x, y) \rightarrow(-y,-x) \) \( (x, y) \rightarrow(-x, y) \) \( (x, y) \rightarrow\{y, x) \)
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To find the coordinates after the \( 90^{\circ} \) counterclockwise rotation, you can use the rule \( (x, y) \rightarrow (-y, x) \). Applying this to the given points: - For \( R(-8, -2) \), we calculate \( R^{\prime} = (-(-2), -8) = (2, -8) \). - For \( S(1, -4) \), we have \( S^{\prime} = (-(-4), 1) = (4, 1) \). - For \( T(-2, -6) \), we find \( T^{\prime} = (-(-6), -2) = (6, -2) \). Now, filling in the blanks, you get: \[ \begin{aligned} R(-8,-2) & \rightarrow R^{\prime}(2, -8) \\ S(1,-4) & \rightarrow S^{\prime}(4, 1) \\ T(-2,-6) & \rightarrow T^{\prime}(6, -2) \end{aligned} \] For part (b), the correct mapping rule that describes the rotation of \( \triangle RST \) to \( \triangle R^{\prime}S^{\prime}T^{\prime} \) is \( (x, y) \rightarrow (-y, x) \). This is the magic behind your shapes spinning around the origin!
