Question

graph △def and its image after a rotation 180 about the origin

Explanation:

Step1: Find coordinates of \( \triangle DEF \)

Assume \( D(-4,-1) \), \( E(-2,3) \), \( F(4,-2) \).

Step2: Apply rotation rule

The rule for a \( 180^{\circ} \) rotation about the origin is \( (x,y)\to(-x,-y) \).
For \( D(-4,-1) \): \( (-4,-1)\to(4,1) \)
For \( E(-2,3) \): \( (-2,3)\to(2,-3) \)
For \( F(4,-2) \): \( (4,-2)\to(-4,2) \)

Step3: Graph the triangles

Plot \( D(-4,-1) \), \( E(-2,3) \), \( F(4,-2) \) for \( \triangle DEF \). Plot \( D'(4,1) \), \( E'(2,-3) \), \( F'(-4,2) \) for the rotated triangle.

Answer:

Graph \( \triangle DEF \) with vertices \( D(-4,-1) \), \( E(-2,3) \), \( F(4,-2) \) and its image after \( 180^{\circ} \) rotation about the origin with vertices \( D'(4,1) \), \( E'(2,-3) \), \( F'(-4,2) \).