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Question Number 67774 by TawaTawa last updated on 31/Aug/19

Answered by MJS last updated on 31/Aug/19

DE=AB=2r  area of semicircle =(π/2)r^2   area of triangle =(1/2)∣AB∣h=rh       with h=rtan x       =r^2 tan x  ⇒  (π/2)r^2 =r^2 tan x  ⇒ x=arctan (π/2) ≈57.52°

$${DE}={AB}=\mathrm{2}{r} \\ $$$$\mathrm{area}\:\mathrm{of}\:\mathrm{semicircle}\:=\frac{\pi}{\mathrm{2}}{r}^{\mathrm{2}} \\ $$$$\mathrm{area}\:\mathrm{of}\:\mathrm{triangle}\:=\frac{\mathrm{1}}{\mathrm{2}}\mid{AB}\mid{h}={rh} \\ $$$$\:\:\:\:\:\mathrm{with}\:{h}={r}\mathrm{tan}\:{x} \\ $$$$\:\:\:\:\:={r}^{\mathrm{2}} \mathrm{tan}\:{x} \\ $$$$\Rightarrow \\ $$$$\frac{\pi}{\mathrm{2}}{r}^{\mathrm{2}} ={r}^{\mathrm{2}} \mathrm{tan}\:{x} \\ $$$$\Rightarrow\:{x}=\mathrm{arctan}\:\frac{\pi}{\mathrm{2}}\:\approx\mathrm{57}.\mathrm{52}° \\ $$

Commented by TawaTawa last updated on 31/Aug/19

God bless you sir

$$\mathrm{God}\:\mathrm{bless}\:\mathrm{you}\:\mathrm{sir} \\ $$

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