Point F lies on segment DE. The coordinates of point $D$ are $(-2,-5)$. The coordinates of point $F$ are $(5,3)$. If $D F: F E=4: 2$, what are the coordinates of point $E$?
Answer (1)
Use the section formula to relate the coordinates of point F to the coordinates of points D and E, given the ratio DF:FE = 2:1.
Set up equations for the x and y coordinates of point F in terms of the coordinates of point E.
Solve the equation for x: 5 = 3 2 x − 2 , which gives x = 8.5 .
Solve the equation for y: 3 = 3 2 y − 5 , which gives y = 7 .
State the coordinates of point E: ( 8.5 , 7 ) .
Explanation
Problem Setup and Given Information We are given that point F lies on segment DE, with D(-2, -5) and F(5, 3). The ratio DF:FE = 4:2, which simplifies to DF:FE = 2:1. We want to find the coordinates of point E.
Applying the Section Formula Let the coordinates of point E be (x, y). We will use the section formula to find the coordinates of point F in terms of the coordinates of points D and E and the ratio DF:FE. The section formula states that if a point F divides the line segment joining points D(x1, y1) and E(x2, y2) in the ratio m:n, then the coordinates of F are given by F = ((mx2 + nx1)/(m+n), (my2 + ny1)/(m+n)).
Setting up Equations for x and y In our case, D(-2, -5), E(x, y), F(5, 3), and the ratio DF:FE = 2:1. Therefore, we can write the coordinates of F as:
5 = 2 + 1 2 x + 1 ∗ ( − 2 ) and 3 = 2 + 1 2 y + 1 ∗ ( − 5 )
Solving for x Now, we solve the equation for x:
5 = 3 2 x − 2
Multiply both sides by 3:
15 = 2 x − 2
Add 2 to both sides:
17 = 2 x
Divide by 2:
x = 2 17 = 8.5
Solving for y Next, we solve the equation for y:
3 = 3 2 y − 5
Multiply both sides by 3:
9 = 2 y − 5
Add 5 to both sides:
14 = 2 y
Divide by 2:
y = 2 14 = 7
Final Answer Therefore, the coordinates of point E are (8.5, 7).
Stating the Solution The coordinates of point E are ( 8.5 , 7 ) .
Examples
In computer graphics, if you need to draw a line between two points and want to extend that line a certain ratio beyond the second point, you can use the section formula to calculate the coordinates of the extended point. For example, if you have a line from point D to point F, and you want to extend it to point E such that DF:FE is a known ratio, this formula helps you find the exact location of point E, ensuring precise graphical rendering.
