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Given-a-circle-with-the-center-at-the-point-O-and-the-radius-of-the-length-R-From-a-point-A-outside-so-that-AO-2R-drawing-two-tangents-AB-and-AC-to-the-circle-B-and-C-are-the-tangency-points-Take-a




Question Number 101937 by 1549442205 last updated on 06/Jul/20
Given a circle with the center at the point O   and the radius of the length R.From a point A outside  so that AO=2R,drawing two tangents AB and AC to the circle  (B and C are the tangency points).Take a arbitrary point M  on smaller arc BC (M differ from B and C)  The tangent pass M cuts AB and AC at Pand Q  respectively.The segments OP and OQ cuts  BC at D and E respectively.  i)Prove that PQ=2DE  ii)Define  the position of M such the   area of the triangle ODE is smallest  and expression it by R
$$\mathrm{Given}\:\mathrm{a}\:\mathrm{circle}\:\mathrm{with}\:\mathrm{the}\:\mathrm{center}\:\mathrm{at}\:\mathrm{the}\:\mathrm{point}\:\mathrm{O}\: \\ $$$$\mathrm{and}\:\mathrm{the}\:\mathrm{radius}\:\mathrm{of}\:\mathrm{the}\:\mathrm{length}\:\mathrm{R}.\mathrm{From}\:\mathrm{a}\:\mathrm{point}\:\mathrm{A}\:\mathrm{outside} \\ $$$$\mathrm{so}\:\mathrm{that}\:\mathrm{AO}=\mathrm{2R},\mathrm{drawing}\:\mathrm{two}\:\mathrm{tangents}\:\mathrm{AB}\:\mathrm{and}\:\mathrm{AC}\:\mathrm{to}\:\mathrm{the}\:\mathrm{circle} \\ $$$$\left(\mathrm{B}\:\mathrm{and}\:\mathrm{C}\:\mathrm{are}\:\mathrm{the}\:\mathrm{tangency}\:\mathrm{points}\right).\mathrm{Take}\:\mathrm{a}\:\mathrm{arbitrary}\:\mathrm{point}\:\mathrm{M} \\ $$$$\mathrm{on}\:\mathrm{smaller}\:\mathrm{arc}\:\mathrm{BC}\:\left(\mathrm{M}\:\mathrm{differ}\:\mathrm{from}\:\mathrm{B}\:\mathrm{and}\:\mathrm{C}\right) \\ $$$$\mathrm{The}\:\mathrm{tangent}\:\mathrm{pass}\:\mathrm{M}\:\mathrm{cuts}\:\mathrm{AB}\:\mathrm{and}\:\mathrm{AC}\:\mathrm{at}\:\mathrm{Pand}\:\mathrm{Q} \\ $$$$\mathrm{respectively}.\mathrm{The}\:\mathrm{segments}\:\mathrm{OP}\:\mathrm{and}\:\mathrm{OQ}\:\mathrm{cuts} \\ $$$$\mathrm{BC}\:\mathrm{at}\:\mathrm{D}\:\mathrm{and}\:\mathrm{E}\:\mathrm{respectively}. \\ $$$$\left.\mathrm{i}\right)\mathrm{Prove}\:\mathrm{that}\:\mathrm{PQ}=\mathrm{2DE} \\ $$$$\left.\mathrm{ii}\right)\mathrm{Define}\:\:\mathrm{the}\:\mathrm{position}\:\mathrm{of}\:\mathrm{M}\:\mathrm{such}\:\mathrm{the}\: \\ $$$$\mathrm{area}\:\mathrm{of}\:\mathrm{the}\:\mathrm{triangle}\:\mathrm{ODE}\:\mathrm{is}\:\mathrm{smallest} \\ $$$$\mathrm{and}\:\mathrm{expression}\:\mathrm{it}\:\mathrm{by}\:\mathrm{R} \\ $$

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