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Category: Geometry

prove-by-contradiction-9-13-3-is-irrational-

Question Number 12822 by fawadalamawan@gmail.com last updated on 02/May/17 $${prove}\:{by}\:{contradiction}\:\mathrm{9}+\mathrm{13}\sqrt{\mathrm{3}\:} \\ $$$${is}\:{irrational} \\ $$ Answered by mrW1 last updated on 03/May/17 $${let}\:{us}\:{assume}\:\mathrm{9}+\mathrm{13}\sqrt{\mathrm{3}}\:{is}\:{rational},.{i}.{e}. \\ $$$${there}\:{exist}\:{integer}\:{numbers}\:{a}\:{and}\:{b},\:{b}\neq\mathrm{0}, \\…

Question-12725

Question Number 12725 by b.e.h.i.8.3.4.1.7@gmail.com last updated on 29/Apr/17 Commented by b.e.h.i.8.3.4.1.7@gmail.com last updated on 29/Apr/17 $${in}\:{triangle}:\:{ABC},{point}\:\:{D}\:{located}\:{on} \\ $$$${triangle}\:{plan},{such}\:{that}: \\ $$$$\angle{ADB}=\angle{BDC}=\angle{CDA}\:{and}: \\ $$$${DH}\parallel{AB}\:,{DG}\parallel{BC}\:,{DF}\parallel{AC}. \\ $$$$\left.\mathrm{1}\right){find}:\:\frac{{DH}}{{AB}}+\frac{{DG}}{{BC}}+\frac{{DF}}{{AC}}\:.…

common-equation-of-conic-sections-ax-2-bxy-cy-2-dx-ey-f-0-if-b-0-we-rotate-tan-2-b-a-c-if-a-c-45-x-x-cos-y-sin-y-x-sin-y-cos-we-now-have-using-x-y-again-instea

Question Number 78246 by MJS last updated on 15/Jan/20 $$\mathrm{common}\:\mathrm{equation}\:\mathrm{of}\:\mathrm{conic}\:\mathrm{sections} \\ $$$${ax}^{\mathrm{2}} +{bxy}+{cy}^{\mathrm{2}} +{dx}+{ey}+{f}=\mathrm{0} \\ $$$$\mathrm{if}\:{b}\neq\mathrm{0}\:\mathrm{we}\:\mathrm{rotate} \\ $$$$\mathrm{tan}\:\mathrm{2}\alpha\:=\frac{{b}}{{a}−{c}}\:\left[\mathrm{if}\:{a}={c}\:\Rightarrow\:\alpha=\mathrm{45}°\right] \\ $$$$\begin{cases}{{x}={x}'\mathrm{cos}\:\alpha\:−{y}'\mathrm{sin}\:\alpha}\\{{y}={x}'\mathrm{sin}\:\alpha\:+{y}'\mathrm{cos}\:\alpha}\end{cases} \\ $$$$\mathrm{we}\:\mathrm{now}\:\mathrm{have}\:\left[\mathrm{using}\:{x},\:{y}\:\mathrm{again}\:\mathrm{instead}\:\mathrm{of}\:{x}',\:{y}'\right] \\ $$$${Ax}^{\mathrm{2}} +{Cy}^{\mathrm{2}}…

we-give-U-1-U-2-U-3-the-terms-of-a-geometric-sequence-Determine-U-1-U-2-U-3-such-that-U-1-U-2-U-3-64-U-1-2-U-2-2-U-3-2-84-

Question Number 12566 by JAZAR last updated on 25/Apr/17 $${we}\:{give}\:{U}_{\mathrm{1}} ,{U}_{\mathrm{2}} ,{U}_{\mathrm{3}} \:{the}\:{terms}\:{of}\:{a}\:{geometric}\:{sequence} \\ $$$$.{Determine}\:{U}_{\mathrm{1}} ,{U}_{\mathrm{2}} ,{U}_{\mathrm{3}} \:{such}\:{that}\:: \\ $$$$ \\ $$$$\begin{cases}{{U}_{\mathrm{1}} .{U}_{\mathrm{2}} .{U}_{\mathrm{3}} =\mathrm{64}}\\{{U}_{\mathrm{1}}…