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Question Number 66746 by John Kaloki Musau last updated on 20/Aug/19

A point T  divides a line AB internally in the ratio 5:2. Given that A is (-4,10) and B is (10,3), find the coordinates of T.

$${A}\:{point}\:{T}\:\:{divides}\:{a}\:{line}\:{AB}\:{internally}\:{in}\:{the}\:{ratio}\:\mathrm{5}:\mathrm{2}.\:{Given}\:{that}\:{A}\:{is}\:\left(-\mathrm{4},\mathrm{10}\right)\:{and}\:{B}\:{is}\:\left(\mathrm{10},\mathrm{3}\right),\:{find}\:{the}\:{coordinates}\:{of}\:{T}. \\ $$

Answered by mr W last updated on 19/Aug/19

x_T =x_A +(5/7)(x_B −x_A )=−4+(5/7)(10+4)=6  y_T =y_A +(5/7)(y_B −y_A )=10+(5/7)(3−10)=5  ⇒T(6,5)

$${x}_{{T}} ={x}_{{A}} +\frac{\mathrm{5}}{\mathrm{7}}\left({x}_{{B}} −{x}_{{A}} \right)=−\mathrm{4}+\frac{\mathrm{5}}{\mathrm{7}}\left(\mathrm{10}+\mathrm{4}\right)=\mathrm{6} \\ $$$${y}_{{T}} ={y}_{{A}} +\frac{\mathrm{5}}{\mathrm{7}}\left({y}_{{B}} −{y}_{{A}} \right)=\mathrm{10}+\frac{\mathrm{5}}{\mathrm{7}}\left(\mathrm{3}−\mathrm{10}\right)=\mathrm{5} \\ $$$$\Rightarrow{T}\left(\mathrm{6},\mathrm{5}\right) \\ $$

Commented by John Kaloki Musau last updated on 19/Aug/19

correct

Answered by Rio Michael last updated on 19/Aug/19

 T(x,y) =( ((μx_1  + λx_2 )/(μ + λ)), ((μy_1  + λy_2 )/(μ + λ)))........internal division  T(x,y) = (((5(−4) + 2(10))/(5+2)) , ((5(10) + 2(3))/(5 + 2)))  T(x,y) = (0, ((56)/7))

$$\:{T}\left({x},{y}\right)\:=\left(\:\frac{\mu{x}_{\mathrm{1}} \:+\:\lambda{x}_{\mathrm{2}} }{\mu\:+\:\lambda},\:\frac{\mu{y}_{\mathrm{1}} \:+\:\lambda{y}_{\mathrm{2}} }{\mu\:+\:\lambda}\right)........{internal}\:{division} \\ $$$${T}\left({x},{y}\right)\:=\:\left(\frac{\mathrm{5}\left(−\mathrm{4}\right)\:+\:\mathrm{2}\left(\mathrm{10}\right)}{\mathrm{5}+\mathrm{2}}\:,\:\frac{\mathrm{5}\left(\mathrm{10}\right)\:+\:\mathrm{2}\left(\mathrm{3}\right)}{\mathrm{5}\:+\:\mathrm{2}}\right) \\ $$$${T}\left({x},{y}\right)\:=\:\left(\mathrm{0},\:\frac{\mathrm{56}}{\mathrm{7}}\right) \\ $$$$ \\ $$

Commented by mr W last updated on 20/Aug/19

please check sir:  it should be  T(x,y) = (((2(−4) + 5(10))/(5+2)) , ((2(10) + 5(3))/(5 + 2)))  T(x,y) = (6, 5)

$${please}\:{check}\:{sir}: \\ $$$${it}\:{should}\:{be} \\ $$$${T}\left({x},{y}\right)\:=\:\left(\frac{\mathrm{2}\left(−\mathrm{4}\right)\:+\:\mathrm{5}\left(\mathrm{10}\right)}{\mathrm{5}+\mathrm{2}}\:,\:\frac{\mathrm{2}\left(\mathrm{10}\right)\:+\:\mathrm{5}\left(\mathrm{3}\right)}{\mathrm{5}\:+\:\mathrm{2}}\right) \\ $$$${T}\left({x},{y}\right)\:=\:\left(\mathrm{6},\:\mathrm{5}\right) \\ $$

Commented by mr W last updated on 20/Aug/19

 T(x,y) =( ((μx_2  + λx_1 )/(μ + λ)), ((μy_2  + λy_1 )/(μ + λ)))........internal division

$$\:{T}\left({x},{y}\right)\:=\left(\:\frac{\mu{x}_{\mathrm{2}} \:+\:\lambda{x}_{\mathrm{1}} }{\mu\:+\:\lambda},\:\frac{\mu{y}_{\mathrm{2}} \:+\:\lambda{y}_{\mathrm{1}} }{\mu\:+\:\lambda}\right)........{internal}\:{division} \\ $$

Commented by Rio Michael last updated on 21/Aug/19

your right sir thanks

$${your}\:{right}\:{sir}\:{thanks} \\ $$

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