Question

1. triangle fgh has vertices f(-3, 4), g(2, 0), and h(-1, -2). graph △fgh and its image after a rotation of 180° about (-3, 1).

Explanation:

Step1: Recall 180° rotation rule

For a point \((x,y)\) rotated 180° about a center \((a,b)\), the new coordinates \((x',y')\) are given by \(x' = 2a - x\), \(y' = 2b - y\). The center here is \((-3,1)\).

Step2: Find image of F(-3,4)

Using the formula: \(x' = 2(-3)-(-3)= -6 + 3=-3\), \(y' = 2(1)-4 = 2 - 4=-2\). So \(F'(-3,-2)\).

Step3: Find image of G(2,0)

\(x' = 2(-3)-2=-6 - 2=-8\), \(y' = 2(1)-0 = 2 - 0 = 2\). So \(G'(-8,2)\).

Step4: Find image of H(-1,-2)

\(x' = 2(-3)-(-1)=-6 + 1=-5\), \(y' = 2(1)-(-2)=2 + 2 = 4\). So \(H'(-5,4)\).

Step5: Graph the triangles

Plot original points \(F(-3,4)\), \(G(2,0)\), \(H(-1,-2)\) and image points \(F'(-3,-2)\), \(G'(-8,2)\), \(H'(-5,4)\). Connect the original points to form \(\triangle FGH\) and the image points to form its 180° rotation about \((-3,1)\).

Answer:

Original triangle vertices: \(F(-3,4)\), \(G(2,0)\), \(H(-1,-2)\); Image vertices after 180° rotation about \((-3,1)\): \(F'(-3,-2)\), \(G'(-8,2)\), \(H'(-5,4)\). Graph by plotting these points and connecting them.