Problem:
In the figure, △ABC has ∠A=45∘ and ∠B=30∘. A line DE, with D on AB and ∠ADE=60∘, divides △ABC into two pieces of equal area. (Note: the figure may not be accurate; perhaps E is on CB instead of AC.) The ratio AD/AB is
Answer Choices:
A. 21
B. 2+22
C. 31
D. 361
E. 4121
Solution:
Point E is on AC, as in the original figure. To show this, we show that if E=C, then △ADE has more than half the area, hence DE is too far right. Indeed, in Figure 1 below, we may assume altitude CF is 1, in which case
Area EAB Area EAD=ABAD=1+31+(1/3)=31>21
Thus we must move DE to the left, as in Figure 2, shrinking the dimensions of △EAD by a factor k so that
Area EAD=21 Area CAB(1/2)k2[1+(1/3)]=(1/4)(1+3),k2=3/2k=43/4
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