Final answer:
Using the formula for dividing a line segment into a given ratio, the coordinates of point T on segment DF partitioned in a 3:1 ratio are (11/2, 7/4). The provided options are incorrect, indicating a possible typo in the question.
Step-by-step explanation:
To find the coordinates of point T that partitions segment DF in a 3:1 ratio, we can use the formula for finding a point that divides a line segment into a given ratio, which is (x, y) = ((mx2 + nx1) / (m + n), (my2 + ny1) / (m + n)), where (x1, y1) and (x2, y2) are the coordinates of the endpoints and m:n is the given ratio. Here, D is (1,4) and F is (7,1), and we want to find T in the ratio of 3:1.
Applying the formula, the x-coordinate of T will be ((3*7 + 1*1) / (3 + 1)) = (21 + 1) / 4 = 22/4 which simplifies to 5.5 or 11/2. The y-coordinate will be ((3*1 + 1*4) / (3 + 1)) = (3 + 4) / 4 = 7/4. So, the coordinates of T are (11/2, 7/4).
Therefore, the correct answer is none of the options provided, suggesting a possible typo in the question. The closest option to the correct answer, which is (11/2, 7/4), would be option b) T=(11/4 , 11/4), but this is still incorrect.