Problem:
In △ABC we have AB=25,BC=39, and AC=42. Points D and E are on AB and AC respectively, with AD=19 and AE=14. What is the ratio of the area of triangle ADE to the area of the quadrilateral BCED?
Answer Choices:
A. 1521266
B. 7519
C. 31
D. 5619
E. 1
Solution:
We have
Area(ABE)Area(ADE)=ABAD=2519 and Area(ABC)Area(ABE)=ACAE=4214=31
so
Area(ADE) Area (ABC)=1925⋅13=1975
and
Area(ADE)Area(BCED)=Area(ADE)Area(ABC)−Area(ADE)=1975−1=1956
Thus
Area(BCED)Area(ADE)=5619
Answer: D.
The problems on this page are the property of the MAA's American Mathematics Competitions
