Final answer:
The point that partitions the directed line segment AB into a 1:3 ratio is found using the section formula and is located at (6.5, 8.5).
Step-by-step explanation:
To find the point that partitions the directed line segment AB into a 1:3 ratio, we will use the section formula. The coordinates of point A are (4, 8) and of point B are (14, 10). First, let's find the x-coordinate of the partitioning point, which we will call point P. To do this, we sum the x-coordinates of A and B, weighed by their respective ratio values, and divide by the sum of the ratio values:
x-coordinate of P = (mx2 + nx1) / (m + n), where m:n is the ratio 1:3, x1 is the x-coordinate of A, and x2 is the x-coordinate of B.
x-coordinate of P = (1*14 + 3*4) / (1 + 3) = (14 + 12) / 4 = 26 / 4 = 6.5
Next, we use a similar approach for the y-coordinate:
y-coordinate of P = (my2 + ny1) / (m + n)
y-coordinate of P = (1*10 + 3*8) / (1 + 3) = (10 + 24) / 4 = 34 / 4 = 8.5
Therefore, the point P that divides the directed line segment AB in a 1:3 ratio is at (6.5, 8.5).