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Question Number 64311 by aseer imad last updated on 16/Jul/19

Commented by Tony Lin last updated on 17/Jul/19

letA′=A−(1,3)=(0,0),C′=C−(1,3)=(4,5)  D′= [((((√2)/2)     0)),((0      ((√2)/2))) ] [((cos(π/4)  −sin(π/4))),((sin(π/4)      cos(π/4))) ] [(4),(5) ]         = [(((1/2)  −(1/2))),(((1/2)      (1/2))) ] [(4),(5) ]          = [((−(1/2))),((    (9/2))) ]  ⇒D=D′+(1,3)=((1/2),((15)/2))  ∵AB^⇀ =DC^⇀   ∴B=(1,3)+((9/2),(1/2))=(((11)/2),(7/2))

$${letA}'={A}−\left(\mathrm{1},\mathrm{3}\right)=\left(\mathrm{0},\mathrm{0}\right),{C}'={C}−\left(\mathrm{1},\mathrm{3}\right)=\left(\mathrm{4},\mathrm{5}\right) \\ $$$${D}'=\begin{bmatrix}{\frac{\sqrt{\mathrm{2}}}{\mathrm{2}}\:\:\:\:\:\mathrm{0}}\\{\mathrm{0}\:\:\:\:\:\:\frac{\sqrt{\mathrm{2}}}{\mathrm{2}}}\end{bmatrix}\begin{bmatrix}{{cos}\frac{\pi}{\mathrm{4}}\:\:−{sin}\frac{\pi}{\mathrm{4}}}\\{{sin}\frac{\pi}{\mathrm{4}}\:\:\:\:\:\:{cos}\frac{\pi}{\mathrm{4}}}\end{bmatrix}\begin{bmatrix}{\mathrm{4}}\\{\mathrm{5}}\end{bmatrix} \\ $$$$\:\:\:\:\:\:\:=\begin{bmatrix}{\frac{\mathrm{1}}{\mathrm{2}}\:\:−\frac{\mathrm{1}}{\mathrm{2}}}\\{\frac{\mathrm{1}}{\mathrm{2}}\:\:\:\:\:\:\frac{\mathrm{1}}{\mathrm{2}}}\end{bmatrix}\begin{bmatrix}{\mathrm{4}}\\{\mathrm{5}}\end{bmatrix} \\ $$$$\:\:\:\:\:\:\:\:=\begin{bmatrix}{−\frac{\mathrm{1}}{\mathrm{2}}}\\{\:\:\:\:\frac{\mathrm{9}}{\mathrm{2}}}\end{bmatrix} \\ $$$$\Rightarrow{D}={D}'+\left(\mathrm{1},\mathrm{3}\right)=\left(\frac{\mathrm{1}}{\mathrm{2}},\frac{\mathrm{15}}{\mathrm{2}}\right) \\ $$$$\because{A}\overset{\rightharpoonup} {{B}}={D}\overset{\rightharpoonup} {{C}} \\ $$$$\therefore{B}=\left(\mathrm{1},\mathrm{3}\right)+\left(\frac{\mathrm{9}}{\mathrm{2}},\frac{\mathrm{1}}{\mathrm{2}}\right)=\left(\frac{\mathrm{11}}{\mathrm{2}},\frac{\mathrm{7}}{\mathrm{2}}\right) \\ $$

Answered by Tanmay chaudhury last updated on 16/Jul/19

AC  (y−3)=((8−3)/(5−1))(x−1)  y−3=(5/4)(x−1)  4y−5x−12+5=0    4y−5x−7=0  is ean AC  let eqn AB (y−3)=m(x−1)  angle betwee AB and AC is (π/4)  tan(π/4)=((m−(5/4))/(1+m×(5/4)))  1+((5m)/4)=m−(5/4)  ((5m)/4)−m=((−5)/4)−1  (m/4)=((−9)/4)   m=−9  AB  (y−3)=((−9)/1)(x−1)  y−3+9x−9=0  y+9x−12=0 ←eqn AB  DC  (y−8)=((−9)/1)(x−5)  y−8+9x−45=0  y+9x−53=0  ←eqn DC  again tan(π/4)=(((5/4)−m)/(1+((5m)/4)))  1+((5m)/4)=(5/4)−m  ((9m)/4)=(1/4)  m=(1/9)  eqn  AD  (y−3)=(1/9)(x−1)  9y−27−x+1=0  9y−x−26=0  ←eqn AD   BC (y−8)=(1/9)(x−5)  9y−72=x−5  9y−x−67=0 ←eqn BC  solve eqn DC  y+9x−53=0 and AD 9y−x−26=0  to find D  y+9(9y−26)−53=0  82y−234−53=0  y=((287)/(82))=(7/2)   so x=9×(7/2)−26  D(((63−52)/2),(7/2))  so D(((11)/2),(7/2))  solveAB y+9x−12=0 BC 9y−x−67=0  y+9(9y−67)−12=0  82y=603+12     y=((615)/(82))=((15)/2)  x=9×((15)/2)−67  x=((135−134)/2)=(1/2)  B((1/2),((15)/2)) so  A(1,3) B((1/2),((15)/2)),C(5,8)  D(((11)/2),(7/2))  pls check   Tanmay 16.07.19

$${AC}\:\:\left({y}−\mathrm{3}\right)=\frac{\mathrm{8}−\mathrm{3}}{\mathrm{5}−\mathrm{1}}\left({x}−\mathrm{1}\right) \\ $$$${y}−\mathrm{3}=\frac{\mathrm{5}}{\mathrm{4}}\left({x}−\mathrm{1}\right) \\ $$$$\mathrm{4}{y}−\mathrm{5}{x}−\mathrm{12}+\mathrm{5}=\mathrm{0}\:\:\:\:\mathrm{4}{y}−\mathrm{5}{x}−\mathrm{7}=\mathrm{0}\:\:{is}\:{ean}\:{AC} \\ $$$${let}\:{eqn}\:{AB}\:\left({y}−\mathrm{3}\right)={m}\left({x}−\mathrm{1}\right) \\ $$$${angle}\:{betwee}\:{AB}\:{and}\:{AC}\:{is}\:\frac{\pi}{\mathrm{4}} \\ $$$${tan}\frac{\pi}{\mathrm{4}}=\frac{{m}−\frac{\mathrm{5}}{\mathrm{4}}}{\mathrm{1}+{m}×\frac{\mathrm{5}}{\mathrm{4}}} \\ $$$$\mathrm{1}+\frac{\mathrm{5}{m}}{\mathrm{4}}={m}−\frac{\mathrm{5}}{\mathrm{4}} \\ $$$$\frac{\mathrm{5}{m}}{\mathrm{4}}−{m}=\frac{−\mathrm{5}}{\mathrm{4}}−\mathrm{1} \\ $$$$\frac{{m}}{\mathrm{4}}=\frac{−\mathrm{9}}{\mathrm{4}}\:\:\:{m}=−\mathrm{9} \\ $$$${AB}\:\:\left({y}−\mathrm{3}\right)=\frac{−\mathrm{9}}{\mathrm{1}}\left({x}−\mathrm{1}\right) \\ $$$${y}−\mathrm{3}+\mathrm{9}{x}−\mathrm{9}=\mathrm{0} \\ $$$${y}+\mathrm{9}{x}−\mathrm{12}=\mathrm{0}\:\leftarrow{eqn}\:{AB} \\ $$$${DC}\:\:\left({y}−\mathrm{8}\right)=\frac{−\mathrm{9}}{\mathrm{1}}\left({x}−\mathrm{5}\right) \\ $$$${y}−\mathrm{8}+\mathrm{9}{x}−\mathrm{45}=\mathrm{0} \\ $$$${y}+\mathrm{9}{x}−\mathrm{53}=\mathrm{0}\:\:\leftarrow{eqn}\:{DC} \\ $$$${again}\:{tan}\frac{\pi}{\mathrm{4}}=\frac{\frac{\mathrm{5}}{\mathrm{4}}−{m}}{\mathrm{1}+\frac{\mathrm{5}{m}}{\mathrm{4}}} \\ $$$$\mathrm{1}+\frac{\mathrm{5}{m}}{\mathrm{4}}=\frac{\mathrm{5}}{\mathrm{4}}−{m} \\ $$$$\frac{\mathrm{9}{m}}{\mathrm{4}}=\frac{\mathrm{1}}{\mathrm{4}}\:\:{m}=\frac{\mathrm{1}}{\mathrm{9}} \\ $$$${eqn}\:\:{AD}\:\:\left({y}−\mathrm{3}\right)=\frac{\mathrm{1}}{\mathrm{9}}\left({x}−\mathrm{1}\right) \\ $$$$\mathrm{9}{y}−\mathrm{27}−{x}+\mathrm{1}=\mathrm{0} \\ $$$$\mathrm{9}{y}−{x}−\mathrm{26}=\mathrm{0}\:\:\leftarrow{eqn}\:{AD}\: \\ $$$${BC}\:\left({y}−\mathrm{8}\right)=\frac{\mathrm{1}}{\mathrm{9}}\left({x}−\mathrm{5}\right) \\ $$$$\mathrm{9}{y}−\mathrm{72}={x}−\mathrm{5} \\ $$$$\mathrm{9}{y}−{x}−\mathrm{67}=\mathrm{0}\:\leftarrow{eqn}\:{BC} \\ $$$${solve}\:{eqn}\:{DC}\:\:{y}+\mathrm{9}{x}−\mathrm{53}=\mathrm{0}\:{and}\:{AD}\:\mathrm{9}{y}−{x}−\mathrm{26}=\mathrm{0} \\ $$$${to}\:{find}\:{D}\:\:{y}+\mathrm{9}\left(\mathrm{9}{y}−\mathrm{26}\right)−\mathrm{53}=\mathrm{0} \\ $$$$\mathrm{82}{y}−\mathrm{234}−\mathrm{53}=\mathrm{0} \\ $$$${y}=\frac{\mathrm{287}}{\mathrm{82}}=\frac{\mathrm{7}}{\mathrm{2}}\:\:\:{so}\:{x}=\mathrm{9}×\frac{\mathrm{7}}{\mathrm{2}}−\mathrm{26} \\ $$$${D}\left(\frac{\mathrm{63}−\mathrm{52}}{\mathrm{2}},\frac{\mathrm{7}}{\mathrm{2}}\right)\:\:{so}\:\boldsymbol{{D}}\left(\frac{\mathrm{11}}{\mathrm{2}},\frac{\mathrm{7}}{\mathrm{2}}\right) \\ $$$${solveAB}\:{y}+\mathrm{9}{x}−\mathrm{12}=\mathrm{0}\:{BC}\:\mathrm{9}{y}−{x}−\mathrm{67}=\mathrm{0} \\ $$$${y}+\mathrm{9}\left(\mathrm{9}{y}−\mathrm{67}\right)−\mathrm{12}=\mathrm{0} \\ $$$$\mathrm{82}{y}=\mathrm{603}+\mathrm{12}\:\:\:\:\:{y}=\frac{\mathrm{615}}{\mathrm{82}}=\frac{\mathrm{15}}{\mathrm{2}} \\ $$$${x}=\mathrm{9}×\frac{\mathrm{15}}{\mathrm{2}}−\mathrm{67} \\ $$$${x}=\frac{\mathrm{135}−\mathrm{134}}{\mathrm{2}}=\frac{\mathrm{1}}{\mathrm{2}} \\ $$$${B}\left(\frac{\mathrm{1}}{\mathrm{2}},\frac{\mathrm{15}}{\mathrm{2}}\right)\:{so} \\ $$$${A}\left(\mathrm{1},\mathrm{3}\right)\:{B}\left(\frac{\mathrm{1}}{\mathrm{2}},\frac{\mathrm{15}}{\mathrm{2}}\right),{C}\left(\mathrm{5},\mathrm{8}\right)\:\:{D}\left(\frac{\mathrm{11}}{\mathrm{2}},\frac{\mathrm{7}}{\mathrm{2}}\right)\:\:{pls}\:{check}\: \\ $$$${Tanmay}\:\mathrm{16}.\mathrm{07}.\mathrm{19} \\ $$$$ \\ $$$$ \\ $$$$ \\ $$

Commented by aseer imad last updated on 16/Jul/19

Thank you very much sir,it is right. Could've avoid the complexity of finding all equations. Instead finding one vertex and use mid point to find the other...

Commented by Tanmay chaudhury last updated on 17/Jul/19

i am going to rechek the steps..

$${i}\:{am}\:{going}\:{to}\:{rechek}\:{the}\:{steps}.. \\ $$

Answered by mr W last updated on 16/Jul/19

M=midpoint of AC=midpoint of BD  M(((1+5)/2),((3+8)/2))=M(3,((11)/2))  Δx_(MB) =Δy_(AM) =((11)/2)−3=(5/2)  ⇒x_B =3+(5/2)=((11)/2)  Δy_(MB) =Δx_(AM) =3−1=2  ⇒y_B =((11)/2)−2=(7/2)  ((x_D +((11)/2))/2)=3  ⇒x_D =6−((11)/2)=(1/2)  ((y_D +(7/2))/2)=((11)/2)  ⇒y_D =11−(7/2)=((15)/2)  ⇒B(((11)/2),(7/2)), D((1/2),((15)/2))

$${M}={midpoint}\:{of}\:{AC}={midpoint}\:{of}\:{BD} \\ $$$${M}\left(\frac{\mathrm{1}+\mathrm{5}}{\mathrm{2}},\frac{\mathrm{3}+\mathrm{8}}{\mathrm{2}}\right)={M}\left(\mathrm{3},\frac{\mathrm{11}}{\mathrm{2}}\right) \\ $$$$\Delta{x}_{{MB}} =\Delta{y}_{{AM}} =\frac{\mathrm{11}}{\mathrm{2}}−\mathrm{3}=\frac{\mathrm{5}}{\mathrm{2}} \\ $$$$\Rightarrow{x}_{{B}} =\mathrm{3}+\frac{\mathrm{5}}{\mathrm{2}}=\frac{\mathrm{11}}{\mathrm{2}} \\ $$$$\Delta{y}_{{MB}} =\Delta{x}_{{AM}} =\mathrm{3}−\mathrm{1}=\mathrm{2} \\ $$$$\Rightarrow{y}_{{B}} =\frac{\mathrm{11}}{\mathrm{2}}−\mathrm{2}=\frac{\mathrm{7}}{\mathrm{2}} \\ $$$$\frac{{x}_{{D}} +\frac{\mathrm{11}}{\mathrm{2}}}{\mathrm{2}}=\mathrm{3} \\ $$$$\Rightarrow{x}_{{D}} =\mathrm{6}−\frac{\mathrm{11}}{\mathrm{2}}=\frac{\mathrm{1}}{\mathrm{2}} \\ $$$$\frac{{y}_{{D}} +\frac{\mathrm{7}}{\mathrm{2}}}{\mathrm{2}}=\frac{\mathrm{11}}{\mathrm{2}} \\ $$$$\Rightarrow{y}_{{D}} =\mathrm{11}−\frac{\mathrm{7}}{\mathrm{2}}=\frac{\mathrm{15}}{\mathrm{2}} \\ $$$$\Rightarrow{B}\left(\frac{\mathrm{11}}{\mathrm{2}},\frac{\mathrm{7}}{\mathrm{2}}\right),\:{D}\left(\frac{\mathrm{1}}{\mathrm{2}},\frac{\mathrm{15}}{\mathrm{2}}\right) \\ $$

Commented by mr W last updated on 17/Jul/19

Answered by ajfour last updated on 17/Jul/19

Commented by ajfour last updated on 17/Jul/19

B(1+4+(1/2) , 3+(1/2)) ⇒ B(((11)/2),(7/2))  D(1−(1/2), 3+(1/2)+4) ⇒ D((1/2),((15)/2)) .

$${B}\left(\mathrm{1}+\mathrm{4}+\frac{\mathrm{1}}{\mathrm{2}}\:,\:\mathrm{3}+\frac{\mathrm{1}}{\mathrm{2}}\right)\:\Rightarrow\:{B}\left(\frac{\mathrm{11}}{\mathrm{2}},\frac{\mathrm{7}}{\mathrm{2}}\right) \\ $$$${D}\left(\mathrm{1}−\frac{\mathrm{1}}{\mathrm{2}},\:\mathrm{3}+\frac{\mathrm{1}}{\mathrm{2}}+\mathrm{4}\right)\:\Rightarrow\:{D}\left(\frac{\mathrm{1}}{\mathrm{2}},\frac{\mathrm{15}}{\mathrm{2}}\right)\:. \\ $$

Commented by aseer imad last updated on 22/Jul/19

Cool! thanks

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