Question

the vertices of \\( \triangle abc \\) are \\( a(-4,4), b(-2,5) \\), and \\( c(-3,2) \\). if \\( \triangle abc \\) is reflected across the line \\( y=-1 \\) to produce the image \\( \triangle a^{prime} b^{prime} c^{prime} \\), find the coordinates of the vertex \\( b^{prime} \\).
the coordinates of \\( b^{prime} \\) after a reflection across the line \\( y=-1 \\) are \\( \square \\).
(type an ordered pair.)

Explanation:

Step1: Calculate the distance between the y - coordinate of point B and the line \(y = - 1\)

The y - coordinate of point \(B(-2,5)\) is \(y = 5\). The distance \(d\) between \(y = 5\) and \(y=-1\) is \(d=\vert5-(-1)\vert=\vert5 + 1\vert=6\).

Step2: Find the y - coordinate of the reflected point \(B'\)

Since we are reflecting across the line \(y=-1\), the y - coordinate of the reflected point \(y_{B'}\) is given by \(y_{B'}=-1-6=-7\) (because the reflection is on the opposite side of the line \(y = - 1\) from the original point). The x - coordinate remains the same.

Answer:

\((-2,-7)\)