SOLUTION: Point J is located at (2,5) and point K is located at (4,19). What are the coordinates of the point that partitions the directed line segment JK in a 3:2 ratio?

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Question 1064977: Point J is located at (2,5) and point K is located at (4,19).
What are the coordinates of the point that partitions the directed line segment JK in a 3:2 ratio?
Found 4 solutions by Fombitz, josgarithmetic, ikleyn, MathTherapy:
Answer by Fombitz(32388) About Me  (Show Source):
You can put this solution on YOUR website!
So the point P is 3/5 of the distance from J to K (starting at J).
Find the x distance,
x=4-2=2
Divide by 5,
dx=2%2F5
Now move 3 x divisions from J,
x%5BP%5D=2%2B3%282%2F5%29=2%2B6%2F5=10%2F5%2B6%2F5=16%2F5
Similarly for y,
y=19-5=14
dy=14%2F5
y%5BP%5D=5%2B14%2F5=25%2F5%2B14%2F5=39%2F5
So the point P is
(16%2F5,39%2F5)

Answer by josgarithmetic(39616) About Me  (Show Source):
You can put this solution on YOUR website!
Line JK is y-5=7%28x-2%29
y=7x-14%2B5
y=7x-9

Using Distance Formula for "the point that partitions the directed line segment JK in a 3:2 ratio",
sqrt%28%28x-2%29%5E2%2B%28y-5%29%5E2%29%2Fsqrt%28%28x-4%29%5E2%2B%28y-19%29%5E2%29=3%2F2

Substitute, simplify, and solve first for x. Choose the solution which would be between J and K, and then evaluate y.

3sqrt%28%28x-4%29%5E2%2B%28y-19%29%5E2%29=2sqrt%28%28x-2%29%5E2%2B%28y-5%29%5E2%29
3sqrt%28%28x-4%29%5E2%2B%287x-9-19%29%5E2%29=2sqrt%28%28x-2%29%5E2%2B%287x-9-5%29%5E2%29
9%28%28x-4%29%5E2%2B%287x-28%29%5E2%29=4%28%28x-2%29%2B%287x-14%29%29
9%28%28x-4%29%5E2%2B7%28x-4%29%5E2%29=4%28%28x-2%29%5E2%2B7%28x-2%29%5E2%29
9%2A8%28x-4%29%5E2=4%2A8%28x-2%29%5E2
72%28x%5E2-8x%2B16%29=32%28x%5E2-128x%2B128%29
.
40x%5E2-448x%2B1024=0
.
highlight_green%285x%5E2-56x%2B128=0%29
(Discriminant, 576=24%5E2)
x=%2856-24%29%2F10
x=32%2F10
highlight%28x=16%2F5%29

Finding y:
y=7x-9
y=7%2816%2F5%29-45%2F5
y=%28112-45%29%2F5
highlight%28y=67%2F5%29

Answer by ikleyn(52775) About Me  (Show Source):
You can put this solution on YOUR website!
.
Ignore the solution by "josgarithmetic".

His approach will lead you to "nowhere".



        "JOSGARITHMETIC"  !!     STOP WRITING NONSENSE  !!!





Answer by MathTherapy(10551) About Me  (Show Source):
You can put this solution on YOUR website!

Point J is located at (2,5) and point K is located at (4,19).
What are the coordinates of the point that partitions the directed line segment JK in a 3:2 ratio?