luong@tonto.DEC (Van Luong Nguyen UHO DTN 264-6560) (07/16/84)
Apology in advance if this is a duplicate posting. Our system crashed while I was trying to mail this, so I am not sure whether it got through the first time yet... >Consider points D,E,F on sides AB,BC,and CA of scalene triangle ABC, with >D,E,F, all distinct from A,B,and C. If area of ABC = 10, AD = 2, DB = 3, and >area of ABE = area of DBEF, what is area of ABE? Terminology: Let's denote the area of a figure by using brackets, eg: ----------- (PQR) = area of triangle PQR Solution: (AEC) = (ABC) - (ABE) -------- Since (ABE)=(DBEF): (AEC) = (ABC) - (DBEF) Therefore: (AEC) = (EFC) + (DAF) (1) But we also have: (AEC) = (EFC)+ (EAF) (2) Comparing (1) and (2) gives: (DAF) = (EAF) (3) Since DAF and EAF share the common base AF, (3) shows that D and E are at the same distance from AF, ie. that lines DE and AC are parallel. As DE and AC are parallel, BE = BD = 3 --- --- --- BC BA 5 But (ABE) = BE therefore (ABE) = (ABC) * BE = 10 * 3 = 6 . ---- --- --- --- (ABC) BC BC 5 NB. Our colleague who posted this puzzle stated that it is from the National Math Exam. Since I have been in the USA only since January 1984, could someone please explain to me what is the National Math Exam? How many years of school do you have to complete before taking this exam? In the school system in France, this would be a medium difficulty problem for Year 9 (ie. 3 years before secondary school graduation). A kid has to do a total of 12 years (5 years in primary school, and 7 in secondary school) before going to college. Van Luong Nguyen, Digital Equipment Corporation, Nashua, NH.