Question

the vertices of \\( \triangle abc \\) are \\( a(-4,4) \\), \\( b(-2,2) \\), and \\( c(-3,2) \\). if \\( \triangle abc \\) is reflected across the line \\( y = 1 \\) to produce the image \\( \triangle abc \\), find the coordinates of the vertex \\( c \\).
the coordinates of \\( c \\) after a reflection across the line \\( y = 1 \\) are \\( \square \\).
(type an ordered pair.)

Explanation:

Step1: Find the distance between point C and the line \(y = 1\)

The \(y -\)coordinate of point \(C(-3,2)\) is \(y = 2\). The distance \(d\) between the point \(C\) and the line \(y=1\) is \(d=\vert2 - 1\vert=1\)

Step2: Calculate the \(y -\)coordinate of \(C'\)

Since we are reflecting across the line \(y = 1\), the \(x -\)coordinate of the point remains the same. For the \(y -\)coordinate, if the original \(y -\)coordinate is \(y_0\) and the line of reflection is \(y = k\), the formula for the reflected \(y -\)coordinate \(y_1\) is \(y_1=2k - y_0\). Here \(k = 1\) and \(y_0=2\), so \(y_1=2\times1-2 = 0\)

Answer:

\((-3,0)\)