what-is-the-area-of-the-largest-square-that-can-be-enclosed-in-a-triangle-with-an-area-of-1-

Question Number 204081 by esmaeil last updated on 05/Feb/24

w h a t i s t h e a r e a o f t h e l a r g e s t s q u a r e

t h a t c a n b e e n c l o s e d i n a t r i a n g l e

w i t h a n a r e a o f 1 ?

Answered by mr W last updated on 06/Feb/24

Commented by mr W last updated on 06/Feb/24

a r e a o f A B C = Δ = 1

Δ = \frac{{a h}_{a}}{2}

\frac{s_{a}}{a} = \frac{h_{a} - s_{a}}{h_{a}}

s_{a} = \frac{{a h}_{a}}{a + h_{a}} = \frac{2 Δ}{a + \frac{2 Δ}{a}} ⩽ \frac{2 Δ}{2 \sqrt{2 Δ}} = \sqrt{\frac{Δ}{2}}

s_{a}^{2} ⩽ \frac{Δ}{2}

\Rightarrow {(s_{a}^{2})}_{m a x} = \frac{Δ}{2} = \frac{1}{2}

w h e n a = \frac{2 Δ}{a}, i . e . a = \sqrt{2 Δ} = h_{a}

s i m i l a r l y {(s_{b}^{2})}_{m a x} = \frac{Δ}{2}, {(s_{c}^{2})}_{m a x} = \frac{Δ}{2} .

\Rightarrow a r e a o f l a r g e s t i n s c r i b e d s q u a r e

i s \frac{Δ}{2} = \frac{1}{2} .

Commented by esmaeil last updated on 06/Feb/24

\underset{―}{\underset{⏟}{⋚}}

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