1 3 x x
（2）①若AC为斜边，则1 x2 (3 x)2，即x2 3x 4 0，无解． ②若AB为斜边，则x2 (3 x)2 1，解得x ③若BC为斜边，则(3 x)2 1 x2，解得x ∴x
5
，满足1 x 2． 3
4
，满足1 x 2． 3
54
或x ． ·············································································································· 9分 33
（3）在△ABC中，作CD AB于D， 设CD h，△ABC的面积为S，则S ①若点D在线段AB上， 则 h2 (3 x)2 h2 x．
1
xh． 2
（第24题-1）
∴(3 x)2 h2 x2 2x h2 1 h2，即x h2 3x 4． ∴x2(1 h2) 9x2 24x 16，即x2h2 8x2 24x 16． ∴S2
412231
． ································· 11分 xh 2x2 6x 4 2(x )2 （≤x 2）
4223
当x
4213
时（满足≤x 2），S2取最大值，从而S取最大值． ························ 13分
2223
②若点D在线段MA上， 则(3 x)2 h2 h2 x．
1
同理可得，S x2h2 2x2 6x 4
4
2
431
， 2(x )2 （1 x≤）
223
易知此时S
2
． 2
（第24题-2）
综合①②得，△ABC的最大面积为
2． ··········································································· 14分 2