Question

what is the angle of rotation about the origin, (0,0), that maps \\(\triangle pqr\\) to \\(\triangle pqr\\)?
choose the correct answer below.
\\(\bigcirc\\) a. \\(360^{\circ}\\)
\\(\bigcirc\\) b. \\(90^{\circ}\\)
\\(\bigcirc\\) c. \\(270^{\circ}\\)
\\(\bigcirc\\) d. \\(180^{\circ}\\)

Explanation:

Step1: Recall Rotation Properties

A \(180^\circ\) rotation about the origin \((x,y)\to(-x,-y)\). We analyze the triangle's vertices.

Step2: Check Vertex Transformations

For a triangle \(PQR\) and its image \(P'Q'R'\) after rotation, if each vertex \((x,y)\) maps to \((-x,-y)\), it's a \(180^\circ\) rotation. A \(90^\circ\) rotation would map \((x,y)\to(-y,x)\) (or other variants), \(270^\circ\) to \((y,-x)\), and \(360^\circ\) is a full rotation (no change). The triangles are opposite in direction, consistent with \(180^\circ\) rotation.

Answer:

D. \(180^\circ\)