Question

select the correct answer.
point f lies on segment de. the coordinates of point d are (-2,-5). the coordinates of point f are (5,3). if df:fe = 4:2, what are the coordinates of point e?
(3,5)
(9,2)
(2 2/3, 1/3)
(8.5,7)

Explanation:

Step1: Use the section formula

If a point \(F(x,y)\) divides the line - segment joining \(D(x_1,y_1)\) and \(E(x_2,y_2)\) in the ratio \(m:n\), then \(x=\frac{mx_2 + nx_1}{m + n}\) and \(y=\frac{my_2+ny_1}{m + n}\). Here, \(D(-2,-5)\), \(F(5,3)\), \(m = 4\), \(n = 2\).
For the \(x\) - coordinate:
We know that \(x=\frac{mx_2+nx_1}{m + n}\), substituting \(x = 5\), \(x_1=-2\), \(m = 4\), \(n = 2\)

$$5=\frac{4x_2+2\times(-2)}{4 + 2}$$

$$5=\frac{4x_2-4}{6}$$

Multiply both sides by \(6\):

$$30=4x_2-4$$

Add \(4\) to both sides:

$$4x_2=34$$

Divide by \(4\):

$$x_2 = 8.5$$

Step2: For the \(y\) - coordinate

We know that \(y=\frac{my_2+ny_1}{m + n}\), substituting \(y = 3\), \(y_1=-5\), \(m = 4\), \(n = 2\)

$$3=\frac{4y_2+2\times(-5)}{4 + 2}$$

$$3=\frac{4y_2-10}{6}$$

Multiply both sides by \(6\):

$$18=4y_2-10$$

Add \(10\) to both sides:

$$4y_2=28$$

Divide by \(4\):

$$y_2 = 7$$

Answer:

\((8.5,7)\)