Points g, k and j are collinear on gj, and gk: gj =3/5. g is located at (-4, 5), k is located at (x, y), and j is located at (6, 0). what are the values of x and y?

  (x, y) = (2, 2)

Step-by-step explanation:

The problem statement tells us ...

  GK : GJ = 3/5 . . . . the given ratio

  5(GK) = 3(GJ) . . . . multiply by 5(GJ)

  5(K-G) = 3(J-G) . . . replace segment lengths with endpoint differences

  5K -5G = 3J -3G . . . eliminate parentheses

  5K = 3J +2G . . . . . add 5G

  K = (3J +2G)/5 . . . . divide by 5

  K = (3(6, 0) +2(-4, 5))/5 = (18 -8, 0+10)/5 . . . . substitute point coordinates

  K = (2, 2)

The values of x and y are both 2.

we have three similar triangles, because each has a right angle and shares an angle.   let's write the angles in order: opposite to short leg, long leg, hypotenuse.

cab similar to bad similar to cbd

or as ratios,

ca: ab: cb = ba: ad: bd = cb: bd: cd

we also know

ac = ad + cd

(ad+cd): ab: cb = ba: ad: bd = cb: bd: cd

(ad+cd)/cb=cb/cd

we have cb=6, ad=5 and seek x=cd.

we reject the negative root and conclude x=4

answer: 4

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