The problem gives the midpoint $M$ of line segment $PQ$ at coordinates $(-5, 2)$, and point $P$ at coordinates $(-7, -2)$. You are asked to find the coordinates of point $Q$.

Midpoint Formula:

The midpoint $M$ of a line segment is given by: $M = \left( \frac{x_1 + x_2}{2}, \frac{y_1 + y_2}{2} \right)$ where $(x_1, y_1)$ and $(x_2, y_2)$ are the coordinates of points $P$ and $Q$, respectively.

Step-by-step solution:

Let the coordinates of point $Q$ be $(x_2, y_2)$.

1. For the $x$-coordinates: $-5 = \frac{-7 + x_2}{2}$ Multiply both sides by 2: $-10 = -7 + x_2$ Add 7 to both sides: $x_2 = -3$

2. For the $y$-coordinates: $2 = \frac{-2 + y_2}{2}$ Multiply both sides by 2: $4 = -2 + y_2$ Add 2 to both sides: $y_2 = 6$

Thus, the coordinates of point $Q$ are $(-3, 6)$.

The correct answer is the last option: $(-3, 6)$.

Would you like a more detailed explanation or have any further questions?

Related questions:

1. How do you find the midpoint of a line segment if both points are known?
2. What is the distance formula between two points in a coordinate plane?
3. How can you determine if three points are collinear using coordinates?
4. What are the properties of a line segment's midpoint in geometry?
5. How do you solve for one endpoint when the midpoint and the other endpoint are given?

Tip:

Always verify your answer by substituting the calculated point back into the midpoint formula to ensure correctness.