To find the coordinates of point [tex]\( E \)[/tex], which partitions the directed line segment from [tex]\( J \)[/tex] to [tex]\( K \)[/tex] in the ratio [tex]\( 1:4 \)[/tex], we will use the section formula for internal division. The coordinates for points [tex]\( J \)[/tex] and [tex]\( K \)[/tex] are given as:
[tex]\( J(-13, -3) \)[/tex] and [tex]\( K(-7, -1) \)[/tex].
The section formula states that if a point [tex]\( E \)[/tex] divides the line segment joining two points [tex]\( J(x_1, y_1) \)[/tex] and [tex]\( K(x_2, y_2) \)[/tex] in the ratio [tex]\( m:n \)[/tex], then the coordinates of [tex]\( E \)[/tex] are given by:
[tex]\[
x = \left(\frac{m}{m+n}\right)(x_2 - x_1) + x_1
\][/tex]
[tex]\[
y = \left(\frac{m}{m+n}\right)(y_2 - y_1) + y_1
\][/tex]
Here, [tex]\( m = 1 \)[/tex] and [tex]\( n = 4 \)[/tex].
Substitute [tex]\( m = 1 \)[/tex], [tex]\( n = 4 \)[/tex], [tex]\( x_1 = -13 \)[/tex], [tex]\( y_1 = -3 \)[/tex], [tex]\( x_2 = -7 \)[/tex], and [tex]\( y_2 = -1 \)[/tex] into the formulas.
First, let's find the [tex]\( x \)[/tex]-coordinate of point [tex]\( E \)[/tex]:
[tex]\[
x = \left(\frac{1}{1+4}\right)(-7 - (-13)) + (-13)
\][/tex]
Simplify the fraction and the expression inside the parentheses:
[tex]\[
x = \left(\frac{1}{5}\right)(-7 + 13) + (-13)
\][/tex]
[tex]\[
x = \left(\frac{1}{5}\right)(6) + (-13)
\][/tex]
[tex]\[
x = \frac{6}{5} - 13
\][/tex]
Convert [tex]\(\frac{6}{5}\)[/tex] to a decimal:
[tex]\[
\frac{6}{5} = 1.2
\][/tex]
So,
[tex]\[
x = 1.2 - 13
\][/tex]
Subtract and get the result for [tex]\( x \)[/tex]:
[tex]\[
x = 1.2 - 13 = -11.8
\][/tex]
Now, let's find the [tex]\( y \)[/tex]-coordinate of point [tex]\( E \)[/tex]:
[tex]\[
y = \left(\frac{1}{1+4}\right)(-1 - (-3)) + (-3)
\][/tex]
Simplify the fraction and the expression inside the parentheses:
[tex]\[
y = \left(\frac{1}{5}\right)(-1 + 3) + (-3)
\][/tex]
[tex]\[
y = \left(\frac{1}{5}\right)(2) + (-3)
\][/tex]
[tex]\[
y = \frac{2}{5} - 3
\][/tex]
Convert [tex]\(\frac{2}{5}\)[/tex] to a decimal:
[tex]\[
\frac{2}{5} = 0.4
\][/tex]
So,
[tex]\[
y = 0.4 - 3
\][/tex]
Subtract and get the result for [tex]\( y \)[/tex]:
[tex]\[
y = 0.4 - 3 = -2.6
\][/tex]
Therefore, the coordinates of point [tex]\( E \)[/tex], which partitions the directed line segment from [tex]\( J \)[/tex] to [tex]\( K \)[/tex] into a ratio of [tex]\( 1:4 \)[/tex], are:
[tex]\[
(-11.8, -2.6)
\][/tex]
