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GeometryQuestion and Answers: Page 105

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A,B,C & D are four distinct points of a circle in such a way that chords AB & CD cut each other inside the circle at the point E. Consequently the circle is divided in four parts (AEC,CEB,BED & DEA). [AEC means the region outlined by AC^(⌢) ,AE^(−) & CE^(−) ] If AE : BE=a:b and CE : DE=c:d, what is ratio between the four parts of the circle?

$$\mathrm{A},\mathrm{B},\mathrm{C}\:\&\:\mathrm{D}\:\mathrm{are}\:\mathrm{four}\:\mathrm{distinct}\:\mathrm{points}\: \\ $$$$\mathrm{of}\:\mathrm{a}\:\mathrm{circle}\:\mathrm{in}\:\mathrm{such}\:\mathrm{a}\:\mathrm{way}\:\mathrm{that}\:\mathrm{chords} \\ $$$$\mathrm{AB}\:\&\:\mathrm{CD}\:\mathrm{cut}\:\mathrm{each}\:\mathrm{other}\:\mathrm{inside} \\ $$$$\mathrm{the}\:\mathrm{circle}\:\mathrm{at}\:\mathrm{the}\:\mathrm{point}\:\mathrm{E}.\:\mathrm{Consequently} \\ $$$$\mathrm{the}\:\mathrm{circle}\:\mathrm{is}\:\mathrm{divided}\:\mathrm{in}\:\mathrm{four}\:\mathrm{parts} \\ $$$$\left(\mathrm{AEC},\mathrm{CEB},\mathrm{BED}\:\&\:\mathrm{DEA}\right). \\ $$$$\left[\mathrm{AEC}\:\mathrm{means}\:\:\mathrm{the}\:\mathrm{region}\:\mathrm{outlined}\:\:\right. \\ $$$$\left.\mathrm{by}\:\overset{\frown} {\mathrm{AC}},\overline {\mathrm{AE}}\:\&\:\overline {\mathrm{CE}}\right] \\ $$$$\mathrm{If}\:\mathrm{AE}\::\:\mathrm{BE}=\mathrm{a}:\mathrm{b}\:\mathrm{and}\:\mathrm{CE}\::\:\mathrm{DE}=\mathrm{c}:\mathrm{d}, \\ $$$$\mathrm{what}\:\mathrm{is}\:\mathrm{ratio}\:\mathrm{between}\:\mathrm{the}\:\mathrm{four} \\ $$$$\mathrm{parts}\:\mathrm{of}\:\mathrm{the}\:\mathrm{circle}? \\ $$

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Prove that the internal bisector of an angle of a triangle divides the opposite sides in the ratio of the sides containing the angle.

$${Prove}\:{that}\:{the}\:{internal}\:{bisector}\:{of}\:{an}\:{angle}\: \\ $$$${of}\:{a}\:{triangle}\:{divides}\:{the}\:{opposite} \\ $$$${sides}\:{in}\:{the}\:{ratio}\:{of}\:{the}\:{sides}\:{containing} \\ $$$${the}\:{angle}. \\ $$

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With a center on a given circle of radius r ,an arc has been drawn in order to divide the circle in two equal (in area) parts. What is the radius of the arc in terms of r (radius of given circle)?

$$\mathrm{With}\:\mathrm{a}\:\mathrm{center}\:\boldsymbol{\mathrm{on}}\:\mathrm{a}\:\mathrm{given}\:\mathrm{circle}\:\mathrm{of} \\ $$$$\mathrm{radius}\:\mathrm{r}\:,\mathrm{an}\:\mathrm{arc}\:\mathrm{has}\:\mathrm{been}\:\mathrm{drawn}\:\mathrm{in}\:\mathrm{order} \\ $$$$\mathrm{to}\:\mathrm{divide}\:\mathrm{the}\:\mathrm{circle}\:\mathrm{in}\:\mathrm{two}\:\mathrm{equal} \\ $$$$\left(\mathrm{in}\:\mathrm{area}\right)\:\mathrm{parts}. \\ $$$$\mathrm{What}\:\mathrm{is}\:\mathrm{the}\:\mathrm{radius}\:\mathrm{of}\:\mathrm{the}\:\mathrm{arc}\:\mathrm{in}\:\mathrm{terms} \\ $$$$\mathrm{of}\:\:\mathrm{r}\:\left(\mathrm{radius}\:\mathrm{of}\:\mathrm{given}\:\mathrm{circle}\right)?\: \\ $$

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Take a point on a given circle as a center and draw an arc which divide the given circle into two equal(in area) regions.Use only Eucledean tools.

$$\mathcal{T}{ake}\:{a}\:{point}\:\boldsymbol{{on}}\:{a}\:{given}\:{circle} \\ $$$${as}\:{a}\:{center}\:{and}\:{draw}\:{an}\:{arc} \\ $$$${which}\:{divide}\:{the}\:{given}\:{circle} \\ $$$${into}\:{two}\:{equal}\left(\mathrm{in}\:\mathrm{area}\right)\:{regions}.{Use}\:{only} \\ $$$${Eucledean}\:{tools}. \\ $$

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ABC is a right tringle.prove it?

$$\mathrm{ABC}\:\mathrm{is}\:\mathrm{a}\:\mathrm{right}\:\mathrm{tringle}.\mathrm{prove}\:\mathrm{it}? \\ $$

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A bus is traveling along a straight road at 100 km/hr and the bus conductor walks at 6 km/hr on the floor of the bus and in the same direction as the bus. Find the speed of the conductor relative to the road and relative to the bus.

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A line passes through A(−3, 0) and B(0, −4). A variable line perpendicular to AB is drawn to cut x and y-axes at M and N. Find the locus of the point of intersection of the lines AN and BM.

$$\mathrm{A}\:\mathrm{line}\:\mathrm{passes}\:\mathrm{through}\:{A}\left(−\mathrm{3},\:\mathrm{0}\right)\:\mathrm{and} \\ $$$${B}\left(\mathrm{0},\:−\mathrm{4}\right).\:\mathrm{A}\:\mathrm{variable}\:\mathrm{line}\:\mathrm{perpendicular} \\ $$$$\mathrm{to}\:{AB}\:\mathrm{is}\:\mathrm{drawn}\:\mathrm{to}\:\mathrm{cut}\:{x}\:\mathrm{and}\:{y}-\mathrm{axes}\:\mathrm{at} \\ $$$${M}\:\mathrm{and}\:{N}.\:\mathrm{Find}\:\mathrm{the}\:\mathrm{locus}\:\mathrm{of}\:\mathrm{the}\:\mathrm{point}\:\mathrm{of} \\ $$$$\mathrm{intersection}\:\mathrm{of}\:\mathrm{the}\:\mathrm{lines}\:{AN}\:\mathrm{and}\:{BM}. \\ $$

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