Geometry

Question Analysis:
This question asks you to find the missing vertex of a quadrilateral so that the shape becomes a parallelogram, and you're expected to use the diagonal bisection property as justification.
The main focus is determining if the diagonals can have the same midpoint, which proves the figure is a parallelogram.

Key Concept Explanation:
In a parallelogram, the diagonals bisect each other.
This means that the midpoint of diagonal AC must equal the midpoint of diagonal BD.

Step-by-Step Solution:
Find the midpoint of diagonal AC:
A is at (2, 1), and C is at (8, 9).
Midpoint = ( (2 + 8) ÷ 2 , (1 + 9) ÷ 2 ) = (5, 5)
Let D be at (x, y). Use midpoint of diagonal BD:
B is at (6, 4), D is unknown
Midpoint of BD = ( (6 + x) ÷ 2 , (4 + y) ÷ 2 )
Set midpoints equal to (5, 5):
(6 + x) ÷ 2 = 5 → 6 + x = 10 → x = 4
(4 + y...