Question

write the coordinates of the vertices after a reflection over the line ( y = -6 ).

Explanation:

Step1: Find the original coordinates

From the graph, we can see that the coordinates of \(A\) is \((1,-2)\), \(B\) is \((6,-2)\) and \(C\) is \((-1,-5)\).

Step2: Use the reflection formula over the line \(y = k\)

The formula for reflecting a point \((x,y)\) over the line \(y=k\) is \((x,2k - y)\). Here \(k=-6\).
For point \(A(1,-2)\):
Substitute \(x = 1\), \(y=-2\) and \(k=-6\) into the formula \((x,2k - y)\)
\(2k-y=2\times(-6)-(-2)=-12 + 2=-10\)
So \(A'\) is \((1,-10)\)
For point \(B(6,-2)\):
Substitute \(x = 6\), \(y=-2\) and \(k=-6\) into the formula \((x,2k - y)\)
\(2k-y=2\times(-6)-(-2)=-12 + 2=-10\)
So \(B'\) is \((6,-10)\)
For point \(C(-1,-5)\):
Substitute \(x=-1\), \(y = -5\) and \(k=-6\) into the formula \((x,2k - y)\)
\(2k-y=2\times(-6)-(-5)=-12 + 5=-7\)
So \(C'\) is \((-1,-7)\)

Answer:

\(A'(1,-10)\), \(B'(6,-10)\), \(C'(-1,-7)\)