Question

write the coordinates of the vertices after a reflection over the line ( x = -1 ).

Explanation:

Step1: Find the distance between the x - coordinate of the point and the line \(x = - 1\)

For a point \((x,y)\) reflected over the line \(x = a\), the formula for the new \(x\) - coordinate is \(x'=2a - x\), and the \(y\) - coordinate remains the same (\(y'=y\)). Here \(a=-1\).
For point \(D(-7,-5)\):
The distance between \(x=-7\) and \(x = - 1\) is \(d=\vert-1-(-7)\vert=\vert-1 + 7\vert = 6\). The new \(x\) - coordinate is \(x'=2\times(-1)-(-7)=-2 + 7=5\), and \(y'=-5\). So \(D'=(5,-5)\)
For point \(E(-7,0)\):
The distance between \(x=-7\) and \(x=-1\) is \(d=\vert-1-(-7)\vert = 6\). The new \(x\) - coordinate is \(x'=2\times(-1)-(-7)=5\), and \(y' = 0\). So \(E'=(5,0)\)
For point \(F(-2,0)\):
The distance between \(x=-2\) and \(x=-1\) is \(d=\vert-1-(-2)\vert=\vert-1 + 2\vert=1\). The new \(x\) - coordinate is \(x'=2\times(-1)-(-2)=-2 + 2=0\), and \(y'=0\). So \(F'=(0,0)\)
For point \(G(-2,-5)\):
The distance between \(x=-2\) and \(x=-1\) is \(d=\vert-1-(-2)\vert = 1\). The new \(x\) - coordinate is \(x'=2\times(-1)-(-2)=0\), and \(y'=-5\). So \(G'=(0,-5)\)

Answer:

\(D'(5,-5)\), \(E'(5,0)\), \(F'(0,0)\), \(G'(0,-5)\)