$\\overline{pq}$ is reflected across the line $x = -3$. the coordinates of the endpoints of the image of…

$\\overline{pq}$ 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 $q$?\n\na $(1, 4)$\nb $(-3, 4)$\nc $(5, 4)$\nd $(-8, 4)$

$\\overline{pq}$ 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 $q$?\n\na $(1, 4)$\nb $(-3, 4)$\nc $(5, 4)$\nd $(-8, 4)$

Answer

Explanation:

Step1: Identify the reflection line and point

The reflection line is $x = -3$ and the image point is $Q'(2, 4)$.

Step2: Determine the y-coordinate of Q

Reflection across a vertical line $x = k$ preserves the y-coordinate. $$y_Q = y_{Q'} = 4$$

Step3: Calculate the x-coordinate of Q

The reflection line $x = k$ is the midpoint of $x_Q$ and $x_{Q'}$. $$\frac{x_Q + x_{Q'}}{2} = k$$

Step4: Substitute values and solve for $x_Q$

Substitute $x_{Q'} = 2$ and $k = -3$ into the formula. $$\frac{x_Q + 2}{2} = -3$$ $$x_Q + 2 = -6$$ $$x_Q = -8$$

Answer:

D. (-8, 4)