Question

Point F lies on overline 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?

Answer 1

The coordinates of point F are (7, 7) and the product of the coordinates is 49.

Step-by-step explanation:

To find the coordinates of point F, we can use the concept of section formula. Let the coordinates of point F be (x, y). According to the section formula, the x-coordinate of F can be found using the formula:

x = (5 * 7 + 1 * 7)/(5 + 1) = 42/6 = 7

Similarly, the y-coordinate of F can be found using the formula:

y = (5 * 9 + (-3) * 1)/(5 + 1) = 42/6 = 7

Therefore, the coordinates of point F are (7, 7).

The product of the coordinates of F is 7 * 7 = 49.

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Answer 2

Point F lies at (6.59, 19.36) and its coordinates' product is 127.58.

Step-by-step explanation:

To calculate point F, we first find the slope between D and E. It's the difference between y-coordinates divided by the difference between x-coordinates: m=(9-(-3))/(7-1)=2. Next, we calculate the total distance from D to E using Euclidean distance: sqrt((7-1)²+(9-(-3))²)=sqrt(36+144)=sqrt(180)=13.4164. So, the distance from D to F is 5/6 times the total distance, i.e., DF=(5/6)*13.4164=11.1803. Substituting the distance to DF in the equation of the line, we find xF=1+11.1803/2=6.59 and yF=-3+2*11.1803=19.36. Thus, the product of the coordinates is 6.59*19.36=127.58.

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