已知Rt△ABC的周长为定值l,求这个三角形面积的最大值.

已知Rt△ABC的周长为定值l,求这个三角形面积的最大值.

答案

解析

:如图,Rt△ABC中,∠C=,

由题意a+b+c=l,                         ①

a2+b2=c2.                                          ②

此三角形面积为S=ab.

由①②可知a+b+=l.

∴2+a+b+=l,

(当且仅当a=b时,等式成立)

即(2+)l.

ab.

S=ab (当且仅当a=b时取等号).

∴三角形面积的最大值为.


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