Question

pq is reflected across the line x = - 3. the coordinates of the endpoints of the image of pq are p(5, 2) and q(2, 4). what are the coordinates of q? a (1, 4) b (-3, 4) c (5, 4) d (-8, 4)

Explanation:

Step1: Find the distance between \(Q'\) and the line \(x = - 3\)

The formula for the distance between a point \((x_1,y_1)\) and the vertical line \(x=a\) is \(d=\vert x_1 - a\vert\).
For the point \(Q'(2,4)\) and the line \(x=-3\), the distance \(d=\vert2-(-3)\vert=\vert2 + 3\vert=5\).

Step2: Calculate the \(x\) - coordinate of \(Q\)

Since reflection across the line \(x=-3\), if the \(x\) - coordinate of \(Q'\) is \(x_{Q'}\) and the \(x\) - coordinate of \(Q\) is \(x_Q\), and the line of reflection is \(x = - 3\), we know that \(\frac{x_Q+x_{Q'}}{2}=-3\).
Substitute \(x_{Q'}=2\) into the formula \(\frac{x_Q + 2}{2}=-3\).
Multiply both sides by \(2\): \(x_Q+2=-6\).
Subtract \(2\) from both sides: \(x_Q=-6 - 2=-8\).
The \(y\) - coordinate of a point \((x,y)\) reflected across a vertical line \(x = a\) remains the same. So \(y_Q=y_{Q'}=4\).

Answer:

D. \((-8,4)\)