Problem:
The area of rectangle ABCD is 72. If point A and the midpoints of BC and CD are joined to form a triangle, the area of that triangle is
Answer Choices:
A. 21
B. 27
C. 30
D. 36
E. 40
Solution:
Three right triangles lie outside △AMN. Their areas are 41​,41​, and 81​ for a total of 85​ of the rectangle. The area of △AMN is 83​(72)=27.
OR
Let the rectangle have sides of 2a and 2b so that 4ab=72 and ab=18. Three right triangles lie outside triangle AMN, and their areas are 21​(2a)(b), 21​(2b)(a), 21​(a)(b), for a total of 25​(ab)=25​(18)=45. The area of triangle AMN is 72−45=27.
Answer: B​.
The problems on this page are the property of the MAA's American Mathematics Competitions
