Answer:
coordinates of point E are (8/3, 1/3)
Explanation:
To find the coordinates of point E, we can use the concept of dividing a line segment in a given ratio. In this case, we know that DF : FE = 4 : 2, which simplifies to 2 : 1.
First, let's find the coordinates of point E. Since DF is twice as long as FE, we can calculate the coordinates of E by moving from F towards D by one-third of the distance between D and F.
To do this, we'll find the difference in x and y coordinates between D and F and then multiply that difference by 1/3.
Difference in x-coordinates: 5 (x-coordinate of F) - (-2) (x-coordinate of D) = 5 + 2 = 7
Difference in y-coordinates: 3 (y-coordinate of F) - (-5) (y-coordinate of D) = 3 + 5 = 8
Now, we'll find one-third of these differences:
One-third of the difference in x-coordinates = (1/3) * 7 = 7/3
One-third of the difference in y-coordinates = (1/3) * 8 = 8/3
Now, we can move one-third of the way from F towards D:
x-coordinate of E = x-coordinate of F - One-third of the difference in x-coordinates = 5 - 7/3 = (15/3) - (7/3) = 8/3
y-coordinate of E = y-coordinate of F - One-third of the difference in y-coordinates = 3 - 8/3 = (9/3) - (8/3) = 1/3