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Category: Geometry

Question-54206

Question Number 54206 by ajfour last updated on 31/Jan/19 Commented by ajfour last updated on 31/Jan/19 $${Find}\:{the}\:{area}\:{of}\:{trapezium}\:{ABCD} \\ $$$${in}\:{terms}\:{of}\:\boldsymbol{{a}}.\:\left({the}\:{red}\:{circles}\:{are}\:{each}\right. \\ $$$${within}\:{a}\:{half}\:{quarter}\:{circle}\:{and}\:{the} \\ $$$$\left.{outer}\:{boundary}\:{is}\:{a}\:{square}\:{of}\:{side}\:\boldsymbol{{a}}\right). \\ $$…

Question-185077

Question Number 185077 by emmanuelson123 last updated on 16/Jan/23 Answered by mahdipoor last updated on 16/Jan/23 $${get}\:{side}\:{of}\:{rectangle}\:{are}\:{A}\:,\:{B}\: \\ $$$$\Delta\mathrm{1}\equiv\Delta\mathrm{2}\equiv\Delta\mathrm{3}\:\:\:\:\Rightarrow \\ $$$${side}\:{of}\:\Delta\mathrm{1}:{A}\:,\:{B}\:,\:\sqrt{{A}^{\mathrm{2}} +{B}^{\mathrm{2}} } \\ $$$${and}\:{r}_{\mathrm{1}}…

Question-185073

Question Number 185073 by emmanuelson123 last updated on 16/Jan/23 Answered by Frix last updated on 16/Jan/23 $$\mathrm{16}°\mathrm{52}'\mathrm{30}''\:=\:\frac{\mathrm{3}\pi}{\mathrm{32}} \\ $$$$\mathrm{sin}\:\frac{\mathrm{3}\pi}{\mathrm{8}}\:=\frac{\sqrt{\mathrm{2}+\sqrt{\mathrm{2}}}}{\mathrm{2}} \\ $$$$\mathrm{sin}\:\frac{{x}}{\mathrm{2}}\:=\sqrt{\frac{\mathrm{1}−\sqrt{\mathrm{1}−\mathrm{sin}^{\mathrm{2}} \:{x}}}{\mathrm{2}}} \\ $$$$\mathrm{sin}\:\frac{\mathrm{3}\pi}{\mathrm{16}}\:=\frac{\sqrt{\mathrm{2}−\sqrt{\mathrm{2}−\sqrt{\mathrm{2}}}}}{\mathrm{2}} \\…

Question-185052

Question Number 185052 by emmanuelson123 last updated on 16/Jan/23 Commented by Frix last updated on 16/Jan/23 $$\forall{x}\in\left[\sqrt{\mathrm{2}},\:+\infty\right):\:−\mathrm{1}\leqslant\frac{\mathrm{1}}{\mathrm{1}−\lfloor{x}^{\mathrm{2}} \rfloor}<\mathrm{0}\:\Rightarrow \\ $$$$\forall{x}\in\left[\sqrt{\mathrm{2}},\:+\infty\right):\:\lfloor\frac{\mathrm{1}}{\mathrm{1}−\lfloor{x}^{\mathrm{2}} \rfloor}\rfloor=−\mathrm{1}\:\Rightarrow \\ $$$${f}\left({x}\right)={x}^{\mathrm{2}} −\mathrm{1};\:\sqrt{\mathrm{2}}\leqslant{x}<+\infty \\…