Final answer:
To find point T which divides the line segment from D (1, 4) to F (7, 1) in a 3:1 ratio, calculate the weighted average of the endpoints' coordinates. The coordinates of T are found to be (5.5, 1.75).
Step-by-step explanation:
To find point T on a segment with endpoints D (1, 4) and F (7, 1) that partitions the segment in a 3:1 ratio, you need to use the concept of dividing a segment into a given ratio, which involves a mixture of the distance formula concepts and algebraic calculation. However, the most appropriate method to find the coordinates of T is by computing a weighted average of the endpoints' coordinates based on the given ratio.
The formula for finding the coordinates of T (x, y) in a ratio of m:n between points D (x1, y1) and F (x2, y2) is given by:
x = (mx2 + nx1) / (m + n)
y = (my2 + ny1) / (m + n)
In this case, the ration m:n is 3:1, so substituting the values we get:
x = (3*7 + 1*1) / (3 + 1) = (21 + 1) / 4 = 22 / 4 = 5.5
y = (3*1 + 1*4) / (3 + 1) = (3 + 4) / 4 = 7 / 4 = 1.75
Therefore, the coordinates of point T are (5.5, 1.75).