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. 2​1​
B. 2+2​2​
C. 3​1​
D. 36​1​
E. 412​1​
Solution:
Point E is on AC, as in the original figure. To show this, we show that if E=C, then â–³ADE 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+3​1+(1/3​)​=3​1​>21​
Thus we must move DE to the left, as in Figure 2, shrinking the dimensions of â–³EAD 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​​
.jpg)
