Problem:
In △ABC,AB=13,AC=5 and BC=12. Points M and N lie on AC and BC, respectively, with CM=CN=4. Points J and K are on AB so that MJ and NK are perpendicular to AB. What is the area of pentagon CMJKN ?
Answer Choices:
A. 15
B. 581
C. 12205
D. 13240
E. 20
Solution:
Because △ABC,△NBK, and △AMJ are similar right triangles whose hypotenuses are in the ratio 13:8:1, their areas are in the ratio 169:64:1.
The area of △ABC is 21(12)(5)=30, so the areas of △NBK and △AMJ are 16964(30) and 1691(30), respectively.
Thus the area of pentagon CMJKN is (1−16964−1691)(30)=240/13.
The problems on this page are the property of the MAA's American Mathematics Competitions
