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

$$whatistheareaofthelargestsquare$$$$thatcanbeenclosedinatriangle$$$$withanareaof1?$$

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

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

$$areaofABC=\mathrm{\Delta }=1$$$$\mathrm{\Delta }=\frac{{ah}_a}{2}$$$$\frac{s_a}{a}=\frac{h_a-s_a}{h_a}$$$$s_a=\frac{{ah}_a}{a+h_a}=\frac{2\mathrm{\Delta }}{a+\frac{2\mathrm{\Delta }}{a}}⩽\frac{2\mathrm{\Delta }}{2\sqrt{2\mathrm{\Delta }}}=\sqrt{\frac{\mathrm{\Delta }}{2}}$$$$s_a^2⩽\frac{\mathrm{\Delta }}{2}$$$$\Rightarrow {\left(s_a^2\right)}_{max}=\frac{\mathrm{\Delta }}{2}=\frac{1}{2}$$$$whena=\frac{2\mathrm{\Delta }}{a},i.e.a=\sqrt{2\mathrm{\Delta }}=h_a$$$$similarly{\left(s_b^2\right)}_{max}=\frac{\mathrm{\Delta }}{2},{\left(s_c^2\right)}_{max}=\frac{\mathrm{\Delta }}{2}.$$$$\Rightarrow areaoflargestinscribedsquare$$$$is\frac{\mathrm{\Delta }}{2}=\frac{1}{2}.$$

Commented by esmaeil last updated on 06/Feb/24

$$\underset{―}{\underbrace{⋚}}$$

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