Question

Point F lies on DE, where D is located at (1, -3) and E is located at (7,9). The ratio of DF to EF is 5:1. What is the product of the coordinates of F? Write your answer as an integer or decimal.

A) -15
B) -21
C) -18
D) 18

Answer 1

The coordinates of point F are (6, 2) and the product of the coordinates is 12.

Step-by-step explanation:

To find the coordinates of point F, we can use the ratio of lengths to find the distance between D and F, and then use that distance to find the coordinates of F.

First, we need to find the distance between D and E using the distance formula:

d = √((x2 - x1)^2 + (y2 - y1)^2)

Plug in the coordinates of D and E to find the distance:

d = √((7 - 1)^2 + (9 - (-3))^2) = √(6^2 + 12^2) = √(36 + 144) = √180 = 6√5

Next, we can find DF and EF using the given ratio:

DF/EF = 5/1

Since the total distance DE is 6√5, DF = (5/6) * 6√5 = 5√5

And EF = (1/6) * 6√5 = √5

Now, we can find the coordinates of F by moving from D towards E by DF:

x-coordinate of F = 1 + 5 = 6

y-coordinate of F = -3 + 5 = 2

Therefore, the coordinates of F are (6, 2).

The product of the coordinates of F is 6 * 2 = 12.