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 formula for the image \((x',y')\) is \(x'=2a - x\) and \(y' = y\) (since reflection over vertical line doesn't change the \(y\)-coordinate). Here, \(a=-3\), and we know the image point \(Q'(2,4)\), we need to find the original point \(Q(x,y)\).

Step2: Apply the reflection formula for \(x\)-coordinate

Using the formula \(x'=2a - x\), substitute \(x' = 2\) and \(a=-3\). So, \(2=2\times(-3)-x\).

Step3: Solve for \(x\)

Simplify the equation: \(2=-6 - x\). Add \(x\) to both sides: \(x + 2=-6\). Then subtract 2 from both sides: \(x=-6 - 2=-8\). The \(y\)-coordinate remains the same, so \(y = 4\). Thus, the coordinates of \(Q\) are \((-8,4)\).

Answer:

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