Question

$\overline{pq}$ is reflected across the line $x = - 3$. the coordinates of the endpoints of the image of $\overline{pq}$ 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 vertical line \(x = - 3\), the \(y\) - coordinate of \(Q\) is the same as the \(y\) - coordinate of \(Q'\) (because reflection across a vertical line does not change the \(y\) - value).
Let the \(x\) - coordinate of \(Q\) be \(x\). Using the property of reflection across the vertical line \(x=-3\): \(\frac{x + 2}{2}=-3\) (the mid - point of \(Q(x,4)\) and \(Q'(2,4)\) lies on the line \(x = - 3\)).
Solving the equation \(x+2=-6\) (multiply both sides of \(\frac{x + 2}{2}=-3\) by 2), we get \(x=-8\).

Answer:

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