To find the coordinates of point $E$, which divides the directed line segment $CD$ three-fourths of the way from $C$ to $D$, we can use the section formula. The section formula states:

\text{If a point P(x, y)$divides the line segment joining$A(x_1, y_1)$and$B(x_2, y_2)$in the ratio$m:n$, then the coordinates of$P are given by:}

$P(x, y) = \left( \frac{mx_2 + nx_1}{m+n}, \frac{my_2 + ny_1}{m+n} \right)$

Step 1: Set Up the Problem

Let's denote the coordinates of points $C$ and $D$ as: $C(x_1, y_1) \quad \text{and} \quad D(x_2, y_2)$

Since point $E$ divides the line segment $CD$ three-fourths of the way from $C$ to $D$, the ratio $m:n$ is $3:1$.

Step 2: Calculate the Coordinates of $E$

Using the section formula: $E(x, y) = \left( \frac{3x_2 + 1x_1}{3+1}, \frac{3y_2 + 1y_1}{3+1} \right)$ $E(x, y) = \left( \frac{3x_2 + x_1}{4}, \frac{3y_2 + y_1}{4} \right)$

Step 3: Plug in Coordinates of $C$ and $D$

Let’s assume the coordinates of $C$ are $(x_1, y_1)$ and $D$ are $(x_2, y_2)$.

$E(x, y) = \left( \frac{3(x_2) + (x_1)}{4}, \frac{3(y_2) + (y_1)}{4} \right)$

Step 4: Example Calculation

If $C$ has coordinates $(1, 2)$ and $D$ has coordinates $(5, 10)$:

$E(x, y) = \left( \frac{3(5) + 1}{4}, \frac{3(10) + 2}{4} \right)$

$E(x, y) = \left( \frac{15 + 1}{4}, \frac{30 + 2}{4} \right)$

$E(x, y) = \left( \frac{16}{4}, \frac{32}{4} \right)$

$E(x, y) = (4, 8)$

Step 5: Plot the Point $E$

I'll now plot the point $E(4, 8)$ on a coordinate plane.

Let's proceed to plot this.