Question

∠MON=90°,点A在OM上,点B在ON上,AB=8cm,A,B沿着OM,ON滑动,问A,B滑到什么位置时,△AOB面

Answer 1

(1)设OA=X,OB=Y,则X²+Y²=AB²=64.

∵4XY=(X+Y)²-(X-Y)²;

4XY=2XY+X²+Y²-(X-Y)²;

4XY=2Xy+64-(X-Y)².

∴2XY=64-(X-Y)².

则当X-Y=0即X=Y时,64-(X-Y)²有最大值.

X²+Y²=2X²=64,则X=4√2,Y=4√2.

即当OA=OB=4√2时,△AOB面积最大.

(2)当X=Y时,X-Y=0,则2XY=64-0²=64.

故2XY最大值也为64,则(1/2)XY最大值为16.

又S△AOB=OA*OB/2=(1/2)XY,所以△AOB面积的最大值为16.