A right angled triangle has fixed hypotenuse measuring h units. What are the measures of its legs, for maximum perimeter P units. Will the area be also maximum, when the perimeter be maximum?


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Question Number 1611 by Rasheed Soomro last updated on 26/Aug/15

$A right angled triangle has fixed hypotenuse measuring$ $h units . What are the measures of its legs,$ $for maximum perimeter P units .$ $Will the area be also maximum, when the perimeter be$ $maximum ?$
Commented by Rasheed Soomro last updated on 28/Aug/15

$L e t t h e m e a s u r e s o f l e g s a r e a a n d b i n g e n e r a l$ $a n d A a n d B i n c a s e p e r i m e t e r i s m a x i m u m :$ $A + B + h ⩾ a + b + h \forall a, b \in R^{+}$ $\Rightarrow A + B ⩾ a + b$ $\Rightarrow {(A + B)}^{2} ⩾ {(a + b)}^{2} [∵ A, B, a, b > 0]$ $\Rightarrow A^{2} + B^{2} + 2 AB ⩾ a^{2} + b^{2} + 2 ab . . . . . . . . . . . . . . . . . . . . . . . (I)$ $B u t A^{2} + B^{2} = a^{2} + b^{2} [= h^{2}] . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . (II)$ $S u b t r a c t i n g (II) f r o m (I) a n d t h e n d i v i d i n g b y 2 :$ $AB ⩾ ab$ $\frac{1}{2} AB ⩾ \frac{1}{2} ab$ $(\begin{matrix} A r e a o f r i g h t a n g l e d t r i a n g l e \\ w i t h f i x e d h y p o t e n u s e h units \\ h a v i n g m a x i m u m p e r i m e t e r \end{matrix}) ⩾ (\begin{matrix} A r e a o f a n y r i g h t a n g l e d \\ t r i a n g l e w i t h \\ f i x e d h y p o t e n u s e h units \end{matrix})$ $This answers second part of the Question .$
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