If δ BCG δ ach, then δ∠BCF =∠ACH.
∠ ECF = 30,∴∠ BCF = 30。
(2) in ∵ RT △ ABC, AC = AB = 2, ∴ AB = 2 √ 2, and the height on the side of AB (set to H) H = √ 2.
∵GH=x,∴BG+HA=2√2-x
Area of y = △ gch = 1/2? x? h=√2x/2
That is, the functional relationship between y and x is: y = √ 2x/2.
When x is the largest, y is the largest.
When CG⊥AB, CG is the smallest and X is the smallest; When CH and CA coincide, CH is the largest and x is the largest.
At this time ∠ CGH =180-(30+45) =105, and x/sin30 = 2/sin 105 is obtained by sine theorem.
x = 2/sin 105×sin 30 = 2/sin 75×sin 30
=2/[(√6+√2)/4]×√3/2
=3√2-√6
That is, when x = 3 √ 2-6, y is the largest.
Y max = √ 2/2× (3 √ 2-√ 6) = 3-√ 3。
Note: SIN 75 = SIN (45+30) = SIN 45 COS30+COS 45 SIN30.
=√2/2×√3/2+√2/2× 1/2
=(√6+√2)/4