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 each vertex and the line \(x = - 1\)

The formula for reflecting a point \((x,y)\) over the line \(x=a\) is \((2a - x,y)\). Here \(a=-1\).
For point \(E(0,-2)\):
The distance between \(x = 0\) and \(x=-1\) is \(|0-(-1)|=1\).
Using the formula \((2\times(-1)-0,y)\), the new \(x\) - coordinate is \(-2\), and \(y\) - coordinate remains \(-2\). So \(E'\) is \((-2,-2)\).
For point \(D(1,0)\):
The distance between \(x = 1\) and \(x=-1\) is \(|1-(-1)| = 2\).
Using the formula \((2\times(-1)-1,y)\), the new \(x\) - coordinate is \(-3\), and \(y\) - coordinate remains \(0\). So \(D'\) is \((-3,0)\).
For point \(B(1,-3)\):
The distance between \(x = 1\) and \(x=-1\) is \(|1-(-1)|=2\).
Using the formula \((2\times(-1)-1,y)\), the new \(x\) - coordinate is \(-3\), and \(y\) - coordinate remains \(-3\). So \(B'\) is \((-3,-3)\).
For point \(C(8,-2)\):
The distance between \(x = 8\) and \(x=-1\) is \(|8-(-1)|=9\).
Using the formula \((2\times(-1)-8,y)\), the new \(x\) - coordinate is \(-10\), and \(y\) - coordinate remains \(-2\). So \(C'\) is \((-10,-2)\).

Answer:

\(E'(-2,-2)\), \(D'(-3,0)\), \(B'(-3,-3)\), \(C'(-10,-2)\)