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APQR = R R(R(AABC)) Given the fact that the coordinate plane below, which single-transformation and the figure provided shown on equation would also be true?

APQR = R R(R(AABC)) Given the fact that the coordinate plane below, which single-transformation and the figure provided shown on equation would also be true?

Advanced Engineering Mathematics
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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Given the fact that \(\triangle PQR = R_{x=1}\left(R_{y=-3}(\triangle ABC)\right)\) and the figure provided shown on the coordinate plane below, which single-transformation equation would also be true? **Diagram Explanation:** - The graph is a Cartesian coordinate plane with both x and y axes ranging from -15 to 15. - A triangle \(\triangle ABC\) is plotted on the grid: - Point \(A\) is positioned at (0, 0). - Point \(B\) is located at (5, 1). - Point \(C\) is at (2, 5). - The triangle is oriented in the first quadrant of the coordinate plane. This setup is designed to analyze transformations in geometry, specifically reflections across given lines \(x = 1\) and \(y = -3\). The task is to determine another transformation that would result in the same positioning as \(\triangle PQR\).
Transcribed Image Text:Given the fact that \(\triangle PQR = R_{x=1}\left(R_{y=-3}(\triangle ABC)\right)\) and the figure provided shown on the coordinate plane below, which single-transformation equation would also be true? **Diagram Explanation:** - The graph is a Cartesian coordinate plane with both x and y axes ranging from -15 to 15. - A triangle \(\triangle ABC\) is plotted on the grid: - Point \(A\) is positioned at (0, 0). - Point \(B\) is located at (5, 1). - Point \(C\) is at (2, 5). - The triangle is oriented in the first quadrant of the coordinate plane. This setup is designed to analyze transformations in geometry, specifically reflections across given lines \(x = 1\) and \(y = -3\). The task is to determine another transformation that would result in the same positioning as \(\triangle PQR\).
The image contains four mathematical expressions related to transformations of triangles: A. \(\triangle PQR = R_{180^\circ, (1, -3)}(\triangle ABC)\) B. \(\triangle PQR = T_{\langle 0, 5 \rangle}(\triangle ABC)\) C. \(\triangle PQR = R_{180^\circ, (-1, 1)}(\triangle ABC)\) D. \(\triangle PQR = T_{\langle 0, -9 \rangle}(\triangle ABC)\) ### Explanation of Symbols: - \(\triangle\) represents a triangle. - \(PQR\) and \(ABC\) are labels for the vertices of the triangles. - \(R_{180^\circ, (x, y)}\) indicates a rotation of 180 degrees around the point \((x, y)\). - \(T_{\langle x, y \rangle}\) indicates a translation by vector \(\langle x, y \rangle\). These transformations describe the relationship between two triangles, \(\triangle ABC\) and \(\triangle PQR\), through either rotation or translation.
Transcribed Image Text:The image contains four mathematical expressions related to transformations of triangles: A. \(\triangle PQR = R_{180^\circ, (1, -3)}(\triangle ABC)\) B. \(\triangle PQR = T_{\langle 0, 5 \rangle}(\triangle ABC)\) C. \(\triangle PQR = R_{180^\circ, (-1, 1)}(\triangle ABC)\) D. \(\triangle PQR = T_{\langle 0, -9 \rangle}(\triangle ABC)\) ### Explanation of Symbols: - \(\triangle\) represents a triangle. - \(PQR\) and \(ABC\) are labels for the vertices of the triangles. - \(R_{180^\circ, (x, y)}\) indicates a rotation of 180 degrees around the point \((x, y)\). - \(T_{\langle x, y \rangle}\) indicates a translation by vector \(\langle x, y \rangle\). These transformations describe the relationship between two triangles, \(\triangle ABC\) and \(\triangle PQR\), through either rotation or translation.
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