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.

Step-by-step explanation:

Consider the horizontal distance from D(1, 4) to F(7, 1).  It is 6 units (to the right from D).  Likewise, the vertical distance from D to F is 3 units (down from D).

We wish to partition each of these two distances in a 3:1 ratio.

Let p be a constant of proportionality.  Then 3p + 1p = 6, or 4p = 6, yielding p = 3/2.  Thus, the horizontal distance (6) is partititioned as follows:

3(3/2) = 9/2 and 1(3/2) = 3/2.  Adding 9/2 to the x-coordinate of D (which is 1) results in the x-coordinate of T being 9/2 + 2/2, or 11/2.

We follow the same pattern to find the y-coordinate of T.

3p + 1p = 3 (vertical distance), or 4p = 3, or p = 3/4.  3(3/4) = 9/4.  The y-coordinate of T is 7 - 9/4, or (28-9)/4, or 19/4.

T is T(11/2, 19/4).