Problem:
A vertical line divides the triangle with vertices (0,0),(1,1) and (9,1) in the xy-plane into two regions of equal area. The equation of the line is x=
Answer Choices:
A. 2.5
B. 3.0
C. 3.5
D. 4.0
E. 4.5
Solution:
In the adjoining figure, ABC is the given triangle and x=a is the dividing line. Since Area △ABC=21​(I)(8)=4, the two regions must each have area 2. Since the portion of △ABC to the left of the vertical line through vertex A has area less than Area △ABF=21​, the line x=a is indeed right of A as shown. Since the equation of the line BC is y=9x​,E is (a, 9a​).
Thus Area △DEC=2=21​(1−9a​)(9−a),or (9−a)2=36. Then 9−a=±6, and a=15 or 3. Since the line x=a must intersect △ABC,x=3.
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