SOLUTION: A point T on a segment with endpoints D (1, 4) and F (7, 1) partitions the segment in a 3:1 ratio. Find T. You must show all work to receive credit.

Question 1205820: A point T on a segment with endpoints D (1, 4) and F (7, 1) partitions the segment in a 3:1 ratio. Find T. You must show all work to receive credit.
Found 3 solutions by MathLover1, ikleyn, greenestamps:
Answer by MathLover1(20850)   (Show Source): You can put this solution on YOUR website!
the coordinates of T are (,) in a ratio of between points D (, ) and F (, ) is given by formula:

in your case, the ratio is , so substituting the values we get:

the coordinates of point are (, )

Answer by ikleyn(52800)   (Show Source): You can put this solution on YOUR website!
.

I want to point out a significant flaw of this formulation.

Under this formulation, there are TWO different ways to make partition of the segment
in the 3:1 ratio.

One partition is 3:1 in direction from D to F,
and totally different partition is 3:1 in direction from F to D.

THEREFORE, an accurate mathematical formulation must indicate the direction of a partition.

Without it, the problem is a garbage for trash can.

Answer by greenestamps(13200)   (Show Source): You can put this solution on YOUR website!

As tutor @ikleyn points out, the statement of the problem is deficient, allowing two different answers.

As tutor MathLover1 shows, you can find the answers (she only found one of them) by plugging numbers into a formula. That tutor loves solving problems by plugging numbers into formulas... but that doesn't help the student learn much.

Let's find one of the answers assuming DT is the longer portion -- that is, DT:TF = 3:1.

Dividing segment DF into two parts with lengths in the ratio 3:1 means T is 3/4 of the way from D to F. Because DF is part of a straight line, the x coordinate of T is 3/4 of the way from the x coordinate of D to the x coordinate of F, and similarly for the y coordinate.

The x coordinate of D is 1; the x coordinate of F is 7; the difference is 6. 3/4 of that difference is 4.5, so the x coordinate of T is 1+4.5 = 5.5.

The y coordinate of D is 4; the y coordinate of F is 1; the difference is -3. 3/4 of that difference is -2.25, so the y coordinate of T is 4-2.25 = 1.75.

ANSWER (1 of them!): T(5.5,1.75)

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