Geometry

Question Analysis:
This is a standard coordinate geometry problem asking for a missing point that will complete a parallelogram.
The main focus is using the diagonal bisection property, which states that the diagonals of a parallelogram bisect each other.

Key Concept Explanation:
If quadrilateral MNPQ is a parallelogram, then the midpoints of both diagonals (MP and NQ) must be the same.
We’ll use this property to determine the coordinates of point Q.

Step-by-Step Solution:
Find the midpoint of diagonal MP:
M = (−1, 3), P = (7, 13)
Midpoint of MP = (−1 + 7)/2, (3 + 13)/2)=(3, 8)
Let Q = (x, y). Find midpoint of diagonal NQ:
N = (5, 7), Q = (x, y)
Midpoint = (5 + x)/2, (7 + y)/2)
Set this equal to the midpoint of MP:
(5 + x)/2 = 3 → 5 + x = 6 → x = 1
(7 + y)/2 = 8 → 7 + y = 16 → y = 9
Therefore, Q = (1, 9)

Common Mistakes:
Choosing the wrong pair of diagonals (e.g., connecti...