Question

write the coordinates of the vertices after a reflection over the line $x = - 5$.

Explanation:

Step1: Find the distance from each point to the line \(x = - 5\)

For a point \((x,y)\), the distance to the line \(x=-5\) is \(d=\vert x-(-5)\vert=\vert x + 5\vert\).
For point \(C(-3,-1)\), \(d=\vert-3 + 5\vert=2\).
For point \(D(-2,-1)\), \(d=\vert-2 + 5\vert=3\).
For point \(E(-2,0)\), \(d=\vert-2 + 5\vert=3\).
For point \(F(-3,0)\), \(d=\vert-3 + 5\vert=2\).

Step2: Calculate the new \(x\) - coordinate

The formula for reflecting a point \((x,y)\) over the line \(x = a\) is \((2a - x,y)\). Here \(a=-5\).
For point \(C(-3,-1)\):
\(x_{new}=2\times(-5)-(-3)=-10 + 3=-7\), \(y_{new}=-1\). So the new point is \(C'(-7,-1)\).
For point \(D(-2,-1)\):
\(x_{new}=2\times(-5)-(-2)=-10 + 2=-8\), \(y_{new}=-1\). So the new point is \(D'(-8,-1)\).
For point \(E(-2,0)\):
\(x_{new}=2\times(-5)-(-2)=-10 + 2=-8\), \(y_{new}=0\). So the new point is \(E'(-8,0)\).
For point \(F(-3,0)\):
\(x_{new}=2\times(-5)-(-3)=-10 + 3=-7\), \(y_{new}=0\). So the new point is \(F'(-7,0)\).

Answer:

\(C'(-7,-1)\), \(D'(-8,-1)\), \(E'(-8,0)\), \(F'(-7,0)\)