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: Recall reflection over vertical line

When a point \((x,y)\) is reflected over the line \(x = a\), the \(y\)-coordinate remains the same, and the \(x\)-coordinate satisfies \(\frac{x + x'}{2}=a\), where \((x',y)\) is the image. Here, \(a=-3\), and we know \(Q'\) is \((2,4)\), so let \(Q=(x,4)\).

Step2: Apply the reflection formula

Using \(\frac{x + 2}{2}=-3\) (since the midpoint of \(Q\) and \(Q'\) lies on \(x = - 3\) and \(y\)-coordinate is same). Multiply both sides by 2: \(x + 2=-6\). Subtract 2: \(x=-6 - 2=-8\). So \(Q=(-8,4)\).

Answer:

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