AB=√(AC? +BC? )=√(6? +8? )= 10
AD=AB/2=5
AP=2t
AQ=AD+ 1t=5+t
The first question:
When AP/AC=AQ/AB, PQ and BC are connected in parallel.
2t/6=(5+t)/ 10
(2/3- 1/5)t= 1
7/ 15t= 1
t= 15/7
When t= 15/7, PQ is parallel to BC.
The second question:
Sina =BC/AB=8/ 10=4/5.
Y = S△APQ = 1/2AP * AQ * Sina = 1/2*2t(5+t)*4/5 = 4/5t? +4t
The third question:
s△ABC = 1/2BC * AC = 1/2 * 8 * 6 = 24
S△APQ = 7/ 15S△ABC
4/5t? +4t = 7/ 15*24 = 56/5
t? +5t - 14 = 0
(t+7)(t-2)=0
t+7≥7
∴t-2=0
t=2
When t=2, the ratio of △APQ to △ABC area is 7: 15.