Question

1. graph $overline{gh}$ with vertices g(5,2), and h(-1,0) and its image after the given composition: rotation: $180^circ$ about the origin reflection: over the line $y = - x$ $g (5,2) \to g (quad,quad ) \to g (quad,quad )$ $h (-1,0) \to h (quad,quad ) \to h (quad,quad )$

Explanation:

Step1: Recall 180° rotation rule

For a point \((x,y)\), a \(180^\circ\) rotation about the origin transforms it to \((-x,-y)\).
For \(G(5,2)\): \(G' = (-5,-2)\)
For \(H(-1,0)\): \(H' = (1,0)\)

Step2: Recall reflection over \(y = -x\) rule

For a point \((x,y)\), reflection over \(y=-x\) transforms it to \((-y,-x)\).
For \(G'(-5,-2)\): \(G'' = (2,5)\) (since \(x=-5,y=-2\), so \(-y = 2\), \(-x = 5\))
For \(H'(1,0)\): \(H'' = (0,-1)\) (since \(x = 1,y = 0\), so \(-y=0\), \(-x=-1\))

Answer:

\(G(5,2)\to G'(-5,-2)\to G''(2,5)\)
\(H(-1,0)\to H'(1,0)\to H''(0,-1)\)