Given directed line segment CD, if point E divides CD three-fourths of the way from C to D, find the coordinates of E, then plot E.

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.



Learn how to find the coordinates of point E that divides the directed line segment CD three-fourths of the way from C to D using the section formula. This problem involves coordinate geometry and section formula application. Suitable for high school students.

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