¶ 2022 AMC8 Problem 18
Problem:
The midpoints of the four sides of a rectangle are:
What is the area of the rectangle?
Answer Choices:
A.
B.
C.
D.
E.
Solution:
The width of the rectangle corresponds to the distancebetween opposite midpoints and . By the Pythagorean Theorem, that distance equals = . Similarly the height of the rectangle corresponds to the distance between midpoints and , which equals . The area of the rectangle is therefore .
The four midpoints form a rhombus with area equal to half the area of the rectangle. The rhombus is a parallelogram with base and height , so its area is . Thus the area of the rectangle is .
Answer: .
The problems on this page are the property of the MAA's American Mathematics Competitions