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GeometryQuestion and Answers: Page 112

Question Number 16067 Answers: 1 Comments: 8

Let d, d′ be two nonparallel lines in the plane and let k > 0. Find the locus of points, the sum of whose distances to d and d′ is equal to k.

$$\mathrm{Let}\:{d},\:{d}'\:\mathrm{be}\:\mathrm{two}\:\mathrm{nonparallel}\:\mathrm{lines}\:\mathrm{in}\:\mathrm{the} \\ $$$$\mathrm{plane}\:\mathrm{and}\:\mathrm{let}\:{k}\:>\:\mathrm{0}.\:\mathrm{Find}\:\mathrm{the}\:\mathrm{locus}\:\mathrm{of} \\ $$$$\mathrm{points},\:\mathrm{the}\:\mathrm{sum}\:\mathrm{of}\:\mathrm{whose}\:\mathrm{distances}\:\mathrm{to} \\ $$$${d}\:\mathrm{and}\:{d}'\:\mathrm{is}\:\mathrm{equal}\:\mathrm{to}\:{k}. \\ $$

Question Number 16066 Answers: 2 Comments: 0

Let ABCD be a convex quadrilateral and let k > 0 be a real number. Find the locus of points M in its interior such that [MAB] + 2[MCD] = k.

$$\mathrm{Let}\:{ABCD}\:\mathrm{be}\:\mathrm{a}\:\mathrm{convex}\:\mathrm{quadrilateral} \\ $$$$\mathrm{and}\:\mathrm{let}\:{k}\:>\:\mathrm{0}\:\mathrm{be}\:\mathrm{a}\:\mathrm{real}\:\mathrm{number}.\:\mathrm{Find} \\ $$$$\mathrm{the}\:\mathrm{locus}\:\mathrm{of}\:\mathrm{points}\:{M}\:\mathrm{in}\:\mathrm{its}\:\mathrm{interior} \\ $$$$\mathrm{such}\:\mathrm{that} \\ $$$$\left[{MAB}\right]\:+\:\mathrm{2}\left[{MCD}\right]\:=\:{k}. \\ $$

Question Number 16065 Answers: 0 Comments: 0

Let ABCD be a convex quadrilateral. Find the locus of points M in its interior such that [MAB] = 2[MCD].

$$\mathrm{Let}\:{ABCD}\:\mathrm{be}\:\mathrm{a}\:\mathrm{convex}\:\mathrm{quadrilateral}. \\ $$$$\mathrm{Find}\:\mathrm{the}\:\mathrm{locus}\:\mathrm{of}\:\mathrm{points}\:{M}\:\mathrm{in}\:\mathrm{its}\:\mathrm{interior} \\ $$$$\mathrm{such}\:\mathrm{that}\:\left[{MAB}\right]\:=\:\mathrm{2}\left[{MCD}\right]. \\ $$

Question Number 16064 Answers: 0 Comments: 8

Let ABCD be a convex quadrilateral and M a point in its interior such that [MAB] = [MBC] = [MCD] = [MDA]. Prove that one of the diagonals of ABCD passes through the midpoint of the other diagonal.

$$\mathrm{Let}\:{ABCD}\:\mathrm{be}\:\mathrm{a}\:\mathrm{convex}\:\mathrm{quadrilateral} \\ $$$$\mathrm{and}\:\mathrm{M}\:\mathrm{a}\:\mathrm{point}\:\mathrm{in}\:\mathrm{its}\:\mathrm{interior}\:\mathrm{such}\:\mathrm{that} \\ $$$$\left[{MAB}\right]\:=\:\left[{MBC}\right]\:=\:\left[{MCD}\right]\:=\:\left[{MDA}\right]. \\ $$$$\mathrm{Prove}\:\mathrm{that}\:\mathrm{one}\:\mathrm{of}\:\mathrm{the}\:\mathrm{diagonals}\:\mathrm{of} \\ $$$${ABCD}\:\mathrm{passes}\:\mathrm{through}\:\mathrm{the}\:\mathrm{midpoint}\:\mathrm{of} \\ $$$$\mathrm{the}\:\mathrm{other}\:\mathrm{diagonal}. \\ $$

Question Number 16053 Answers: 0 Comments: 1

number of positive integers a and b and c satisfying a^b^c b^c^a c^a^b =5abc

$${number}\:{of}\:{positive}\:{integers}\:\boldsymbol{{a}}\:\boldsymbol{{and}}\:\boldsymbol{{b}}\:\boldsymbol{{and}}\:\boldsymbol{{c}}\:\boldsymbol{{satisfying}} \\ $$$$\boldsymbol{{a}}^{\boldsymbol{{b}}^{\boldsymbol{{c}}} } \boldsymbol{{b}}^{\boldsymbol{{c}}^{\boldsymbol{{a}}} } \boldsymbol{{c}}^{\boldsymbol{{a}}^{\boldsymbol{{b}}} } =\mathrm{5}\boldsymbol{{abc}} \\ $$

Question Number 16014 Answers: 1 Comments: 0

A cirlce is drawn to touch the sides of a triangle whose sides are 12cm,10cm,and 9cm. Find the radius of the circle.

$$\mathrm{A}\:\mathrm{cirlce}\:\mathrm{is}\:\mathrm{drawn}\:\mathrm{to}\:\mathrm{touch}\:\mathrm{the} \\ $$$$\mathrm{sides}\:\mathrm{of}\:\mathrm{a}\:\mathrm{triangle}\:\mathrm{whose}\:\mathrm{sides}\:\mathrm{are} \\ $$$$\mathrm{12cm},\mathrm{10cm},\mathrm{and}\:\mathrm{9cm}.\:\mathrm{Find}\:\mathrm{the}\: \\ $$$$\mathrm{radius}\:\mathrm{of}\:\mathrm{the}\:\mathrm{circle}. \\ $$

Question Number 15987 Answers: 1 Comments: 1

Question Number 15982 Answers: 0 Comments: 3

Question Number 15969 Answers: 0 Comments: 17

Question Number 15919 Answers: 0 Comments: 1

multiply 3x+4y+5x−8y

$${multiply}\:\mathrm{3}{x}+\mathrm{4}{y}+\mathrm{5}{x}−\mathrm{8}{y} \\ $$

Question Number 15917 Answers: 1 Comments: 1

Question Number 15908 Answers: 1 Comments: 7

Question Number 15904 Answers: 2 Comments: 1

Question Number 15784 Answers: 1 Comments: 1

Question Number 15782 Answers: 1 Comments: 1

Question Number 15700 Answers: 1 Comments: 1

Question Number 15699 Answers: 1 Comments: 0

3+5x=58

$$\mathrm{3}+\mathrm{5x}=\mathrm{58} \\ $$

Question Number 15672 Answers: 1 Comments: 0

solve the equation: {5^x } +5x=140 please show workings.....

$$\:{solve}\:{the}\:{equation}: \\ $$$$\:\:\left\{\mathrm{5}^{{x}} \right\}\:+\mathrm{5}{x}=\mathrm{140} \\ $$$${please}\:{show}\:{workings}..... \\ $$

Question Number 15642 Answers: 1 Comments: 0

Question Number 15572 Answers: 2 Comments: 1

Question Number 15567 Answers: 2 Comments: 3

Question Number 16613 Answers: 1 Comments: 2

Question Number 15440 Answers: 3 Comments: 9

Question Number 15434 Answers: 1 Comments: 8

Question Number 15318 Answers: 2 Comments: 6

Question Number 15175 Answers: 2 Comments: 1

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