Find the exact length of line segment \( \overline{P Q} \) given \( P(x, y)=(18,-4) \) and the midpoint of \( \overline{P Q} \) is \( M(x, y)=(-4,12) \). \( \overline{P Q}=\square \)

To find the length of the line segment \( \overline{PQ} \), we first need to determine the coordinates of point \( Q \). The midpoint \( M \) of segment \( \overline{PQ} \) is found using the midpoint formula: \[ M\left( \frac{x_P + x_Q}{2}, \frac{y_P + y_Q}{2} \right) = M(x, y) \] This allows us to set up equations based on the coordinates of \( P \) and \( M \): 1. For the x-coordinates: \[ \frac{18 + x_Q}{2} = -4 \] Multiplying both sides by 2: \[ 18 + x_Q = -8 \] Solving for \( x_Q \): \[ x_Q = -8 - 18 = -26 \] 2. For the y-coordinates: \[ \frac{-4 + y_Q}{2} = 12 \] Multiplying both sides by 2: \[ -4 + y_Q = 24 \] Solving for \( y_Q \): \[ y_Q = 24 + 4 = 28 \] Now we have the coordinates of point \( Q \): \[ Q(-26, 28) \] Next, we find the length of line segment \( \overline{PQ} \) using the distance formula: \[ d = \sqrt{(x_Q - x_P)^2 + (y_Q - y_P)^2} \] Substituting in our points \( P(18, -4) \) and \( Q(-26, 28) \): \[ d = \sqrt{((-26) - 18)^2 + (28 - (-4))^2} \] \[ = \sqrt{(-44)^2 + (32)^2} \] \[ = \sqrt{1936 + 1024} \] \[ = \sqrt{2960} \] We can simplify \( \sqrt{2960} \) as follows: \[ 2960 = 16 \times 185 \] \[ \sqrt{2960} = 4\sqrt{185} \] Thus, the exact length of line segment \( \overline{PQ} \) is: \[ \overline{PQ} = 4\sqrt{185} \]