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Co-ordinate geometry

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Find the ratio in which the point(-3,k) divides the line segment joining the points (-5,-4) and (-2,3). Also find the value of k.

i'm getting the value of k as 5 using the formula $$\sqrt{}(x2-x1)2 +(y2-y1)2$$ i hope u understood the formula it's (x2-x1)² and same with y and root sqr . but i'm getting the ratio as fraction. please solve this so i can see what mistake i made.

SARAHann  Apr 1, 2017
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You probably will not like my answe much Sara because I do not do questions like this by formula.

I can never remember formulas that I do not think I really need.  It is some weird kind of mental block

Find the ratio in which the point(-3,k) divides the line segment joining the points (-5,-4) and (-2,3). Also find the value of k.

Look at the x values first

-2--5=3

-3--5=2

so the dividing point  is 2/3 of the way starting from the first end point.           The ratio is 2:3

so the k value must also be 2/3 of the way, starting from the first end point

Now the y difference in y values must be in the same ratio.

3--4=7

k--4 = k+4

$$\frac{k+4}{7}=\frac{2}{3}\\ 3k+12=14\\ 3k=2\\ k=\frac{2}{3}$$

Melody  Apr 1, 2017
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Mel, I did'nt understand it. I need it with the formula i've given

SARAHann  Apr 1, 2017
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Maybe you understand CPhill's explanation better ?

It is good that you said you did not understand.  I appraciate you being honest :))

Melody  Apr 1, 2017
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Thanks, Melody.....here's another way....

Slope of the line  = [ 3 - - 4 ] / [ - 2 - - 5]  =  7 / 3

And the equation of the line is

y = ( 7 / 3) (x - -2) + 3

y = (7 / 3) (x + 2) + 3      and when x  = -3, we can find k as

k = ( 7 / 3) ( -3 + 2)  + 3

k = ( 7 / 3) (-1)  + 3   =   2/3    so.....the point is ( -3, 2/3 )

And the ratio is given by

[Distance from (-5, -4) to ( - 3 , 2/3) ] /  [ Distance from ( -3, 2/3) to ( -2 , 3) ]

Distance from (-5, -4) to (-3, 2/3)   =  sqrt [ ( -3 - - 5)^2 + ( - 4 -2 /3)^2 ]

sqrt [ 4   + (-14/3)^2 ] =  (2/3)sqrt(58)

Distance from ( -3, 2/3) to ( -2 , 3) =  sqrt [ (-2 - - 3)^2 + (3 - 2/3)^2 ]   =

sqrt [ 1  + (7/3)^2 ] = (1/3) sqrt(58)

So......(-3, 2/3)   divides the line in the ratio of :

[(2/3)  sqrt (58)] / [ (1/3) sqrt (58) ]   =

[2/3] / [ 1/3]  =  2 / 1  =    2 :  1

CPhill  Apr 1, 2017

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