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Question Number 45649 by hassentimol last updated on 15/Oct/18

Let consider A(3,5), B(6,4) and C(3,−2), d : x−5y+7=0 Consider a dot D such as ABCD is a trapezium and (AD) and (BC) are parallel lines. Q1) Give the equation of the line which contains the point D. Q2) Considering that the trapezium should be convex, are all the points D of the lines correct for the Trapezium ? Which ones are ? Give a proof. I have some difficulties to anzwer the question n°2, could someone please, help me. Thank you. H.T.

$$\mathrm{Let}\:\mathrm{consider}\:\mathrm{A}\left(\mathrm{3},\mathrm{5}\right),\:\mathrm{B}\left(\mathrm{6},\mathrm{4}\right)\:\mathrm{and}\:\mathrm{C}\left(\mathrm{3},−\mathrm{2}\right), \\ $$$$\mathrm{d}\::\:{x}−\mathrm{5}{y}+\mathrm{7}=\mathrm{0} \\ $$$$ \\ $$$$\mathrm{Consider}\:\mathrm{a}\:\mathrm{dot}\:\mathrm{D}\:\mathrm{such}\:\mathrm{as}\:\mathrm{ABCD}\:\mathrm{is}\:\mathrm{a}\:\mathrm{trapezium} \\ $$$$\mathrm{and}\:\left(\mathrm{AD}\right)\:\mathrm{and}\:\left(\mathrm{BC}\right)\:\mathrm{are}\:\mathrm{parallel}\:\mathrm{lines}. \\ $$$$ \\ $$$$\left.\mathrm{Q1}\right)\:\:\:\mathrm{Give}\:\mathrm{the}\:\mathrm{equation}\:\mathrm{of}\:\mathrm{the}\:\mathrm{line}\:\mathrm{which} \\ $$$$\mathrm{contains}\:\mathrm{the}\:\mathrm{point}\:\mathrm{D}. \\ $$$$ \\ $$$$\left.\mathrm{Q2}\right)\:\:\:\mathrm{Considering}\:\mathrm{that}\:\mathrm{the}\:\mathrm{trapezium} \\ $$$$\mathrm{should}\:\mathrm{be}\:\mathrm{convex},\:\mathrm{are}\:\mathrm{all}\:\mathrm{the}\:\mathrm{points}\:\mathrm{D}\:\mathrm{of} \\ $$$$\mathrm{the}\:\mathrm{lines}\:\mathrm{correct}\:\mathrm{for}\:\mathrm{the}\:\mathrm{Trapezium}\:? \\ $$$$\mathrm{Which}\:\mathrm{ones}\:\mathrm{are}\:? \\ $$$$\mathrm{Give}\:\mathrm{a}\:\mathrm{proof}. \\ $$$$ \\ $$$$ \\ $$$$\mathrm{I}\:\mathrm{have}\:\mathrm{some}\:\mathrm{difficulties}\:\mathrm{to}\:\mathrm{anzwer}\:\mathrm{the} \\ $$$$\mathrm{question}\:\mathrm{n}°\mathrm{2},\:\mathrm{could}\:\mathrm{someone}\:\mathrm{please}, \\ $$$$\mathrm{help}\:\mathrm{me}. \\ $$$$ \\ $$$$\mathrm{Thank}\:\mathrm{you}. \\ $$$$ \\ $$$$\mathrm{H}.\mathrm{T}. \\ $$

Commented by hassentimol last updated on 15/Oct/18

Could you help me for Question n°2 ? How may I proove ?

$$ \\ $$$$ \\ $$$$ \\ $$$$ \\ $$$$ \\ $$$$\mathrm{Could}\:\mathrm{you}\:\mathrm{help}\:\mathrm{me}\:\mathrm{for}\:\mathrm{Question}\:\mathrm{n}°\mathrm{2}\:? \\ $$$$\mathrm{How}\:\mathrm{may}\:\mathrm{I}\:\mathrm{proove}\:? \\ $$

Answered by tanmay.chaudhury50@gmail.com last updated on 15/Oct/18

Slope of BC=slope of AD=((−2−4)/(3−6))=((−6)/(−3))=2 eqn AD (y−5)=2(x−3) y−2x−5+6=0 y−2x+1=0 Point D lies/on y−2x+1=0

$${Slope}\:{of}\:{BC}={slope}\:{of}\:{AD}=\frac{−\mathrm{2}−\mathrm{4}}{\mathrm{3}−\mathrm{6}}=\frac{−\mathrm{6}}{−\mathrm{3}}=\mathrm{2} \\ $$$${eqn}\:{AD}\:\left({y}−\mathrm{5}\right)=\mathrm{2}\left({x}−\mathrm{3}\right) \\ $$$${y}−\mathrm{2}{x}−\mathrm{5}+\mathrm{6}=\mathrm{0}\:\:\:\:{y}−\mathrm{2}{x}+\mathrm{1}=\mathrm{0}\:\:\:{Point}\:{D}\:{lies}/{on} \\ $$$${y}−\mathrm{2}{x}+\mathrm{1}=\mathrm{0} \\ $$

Commented by hassentimol last updated on 15/Oct/18

Thank you very much sir ! Could you help me for question n°2; I have understood that the dot D should be like : x_D < 3, because the lines of the trapezium must not be crossed (as they are convex), however, how could I give a proof to the answer ?

$$ \\ $$$$ \\ $$$${Thank}\:{you}\:{very}\:{much}\:{sir}\:! \\ $$$$ \\ $$$$\mathrm{Could}\:\mathrm{you}\:\mathrm{help}\:\mathrm{me}\:\mathrm{for}\:\mathrm{question}\:\mathrm{n}°\mathrm{2};\:\mathrm{I}\:\mathrm{have} \\ $$$$\mathrm{understood}\:\mathrm{that}\:\mathrm{the}\:\mathrm{dot}\:\mathrm{D}\:\mathrm{should}\:\mathrm{be}\:\mathrm{like}\:: \\ $$$${x}_{{D}} \:<\:\mathrm{3},\:\mathrm{because}\:\mathrm{the}\:\mathrm{lines}\:\mathrm{of}\:\mathrm{the}\:\mathrm{trapezium} \\ $$$$\mathrm{must}\:\mathrm{not}\:\mathrm{be}\:\mathrm{crossed}\:\left(\mathrm{as}\:\mathrm{they}\:\mathrm{are}\:\mathrm{convex}\right),\:\mathrm{however},\:\mathrm{how}\:\mathrm{could}\:\mathrm{I} \\ $$$$\mathrm{give}\:\mathrm{a}\:\mathrm{proof}\:\mathrm{to}\:\mathrm{the}\:\mathrm{answer}\:? \\ $$$$ \\ $$

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