Final answer:
The area of rhombus ABCD with vertices A(1,2), B(4,6), C(7,2), and D(4,-2) is 24 square units.
Step-by-step explanation:
The area of rhombus ABCD can be determined by calculating the product of the lengths of its diagonals divided by 2. The coordinates A(1,2), B(4,6), C(7,2), and D(4,-2) allow us to find the lengths of the diagonals AC and BD. The diagonal AC is a straight line from A to C and its length can be found using the distance formula:
AC = √[(7-1)^2 + (2-2)^2] = √[36+0] = 6 units.
The diagonal BD is a straight line from B to D and its length can be found similarly:
BD = √[(4-4)^2 + (6+2)^2] = √[64] = 8 units.
Now, the area of the rhombus is half the product of the lengths of the diagonals:
Area = ½ × AC × BD = ½ × 6 × 8 = 24 square units.