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The problem asks to graph point \( F \) on line segment \( DE \), such that the ratio of \( DF \) to \( EF \) is \( 6:1 \). This means \( F \) divides \( DE \) in the ratio of 6 to 1, with \( F \) being closer to \( E \).

Here’s how we can solve it:

### Step 1: Identify the coordinates of points \( D \) and \( E \)
From the graph:
- \( D = (-8, -6) \)
- \( E = (4, 10) \)

### Step 2: Use the section formula
For a line segment divided in the ratio \( m:n \), the coordinates of the point \( F(x, y) \) dividing the segment are given by:
\[
F(x, y) = \left( \frac{m \cdot x_2 + n \cdot x_1}{m+n}, \frac{m \cdot y_2 + n \cdot y_1}{m+n} \right)
\]
Where \( (x_1, y_1) \) and \( (x_2, y_2) \) are the coordinates of points \( D \) and \( E \), and \( m:n \) is the ratio.

Given \( DF:EF = 6:1 \), we have \( m = 6 \) and \( n = 1 \). Now, substitute the coordinates of \( D(-8, -6) \) and \( E(4, 10) \) into the formula.

### Step 3: Calculate the coordinates of \( F \)
Let’s compute the \( x \)-coordinate and \( y \)-coordinate separately.

1. **\( x \)-coordinate of \( F \):**
\[
x_F = \frac{6 \cdot 4 + 1 \cdot (-8)}{6+1} = \frac{24 - 8}{7} = \frac{16}{7} \approx 2.29
\]

2. **\( y \)-coordinate of \( F \):**
\[
y_F = \frac{6 \cdot 10 + 1 \cdot (-6)}{6+1} = \frac{60 - 6}{7} = \frac{54}{7} \approx 7.71
\]

Thus, \( F \) is approximately at \( \left( 2.29, 7.71 \right) \).

### Step 4: Plot point \( F \)
You can plot \( F \) at these coordinates, and it should be very close to the line segment between \( D \) and \( E \).

Would you like further clarification or additional details?

### Related Questions:
1. What is the general formula for dividing a line segment in a given ratio?
2. How does the ratio affect the position of the point dividing the segment?
3. What are some real-world applications of the section formula?
4. How can we extend this idea to three-dimensional geometry?
5. How can the section formula be used to find the midpoint of a segment?

### Tip:
When working with ratios in geometry, always ensure you have the correct order of the points and ratio to avoid misplacing the dividing point!

Point F lies on DE such that DF:EF is 6:1. Graph F.

Learn how to graph point F on line segment DE when DF:EF is 6:1. This solution covers key geometry concepts, the section formula, and step-by-step calculations. Suitable for students in Grades 8-10.

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