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-

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 \dots \dots \dots \dots \dots \dots \dots . . (I)

B u t A^{2} + B^{2} = a^{2} + b^{2} [= h^{2}] \dots \dots \dots \dots \dots \dots \dots \dots \dots \dots . . (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 .