The problem is asking for the coordinates of the other endpoint $P(x_1, y_1)$ of line segment $PQ$, given the midpoint $M(71, 55)$ and the other endpoint $Q(96, 58)$.

We can use the midpoint formula:

$M = \left( \frac{x_1 + x_2}{2}, \frac{y_1 + y_2}{2} \right)$

where $M(x_m, y_m)$ is the midpoint, and $P(x_1, y_1)$ and $Q(x_2, y_2)$ are the endpoints.

Given:

• $M(71, 55)$
• $Q(96, 58)$ (so $x_2 = 96$ and $y_2 = 58$)

We need to find $P(x_1, y_1)$.

Step 1: Solve for $x_1$ (the x-coordinate of P)

$71 = \frac{x_1 + 96}{2}$ Multiply both sides by 2: $142 = x_1 + 96$ Solve for $x_1$: $x_1 = 142 - 96 = 46$

Step 2: Solve for $y_1$ (the y-coordinate of P)

$55 = \frac{y_1 + 58}{2}$ Multiply both sides by 2: $110 = y_1 + 58$ Solve for $y_1$: $y_1 = 110 - 58 = 52$

Final Answer:

The coordinates of point $P$ are $P(46, 52)$.

Would you like further details or have any questions?

Here are 5 related questions to deepen your understanding:

1. How can you find the distance between the two endpoints $P$ and $Q$?
2. How would the midpoint formula change in 3D space?
3. Can you apply the midpoint formula if you know the midpoint and both endpoints?
4. How would the solution change if the midpoint were instead an average of multiple points?
5. What if the midpoint is located outside of a regular Cartesian coordinate system (e.g., on a polar grid)?

Tip: When solving for missing coordinates using the midpoint formula, make sure to isolate one variable at a time by using basic algebraic manipulation.