https://file.aiquickdraw.com/mathbot/file/17254901140242q03icgx.jpg

To solve the problem of finding point \( G \) on the line segment \( EF \) such that \( EG:FG = 1:3 \), we can follow these steps:

### Step 1: Determine the coordinates of points \( E \) and \( F \)
- Point \( E \) has coordinates \((1, -8)\).
- Point \( F \) has coordinates \((6, 4)\).

### Step 2: Apply the section formula
Since \( G \) divides \( EF \) in the ratio \( 1:3 \), we use the section formula to find the coordinates of \( G \). The section formula for a point dividing a line segment internally in the ratio \( m:n \) is given by:

\[
G\left(\frac{mx_2 + nx_1}{m+n}, \frac{my_2 + ny_1}{m+n}\right)
\]

Here:
- \( m = 1 \)
- \( n = 3 \)
- \( (x_1, y_1) = (1, -8) \) (coordinates of \( E \))
- \( (x_2, y_2) = (6, 4) \) (coordinates of \( F \))

### Step 3: Calculate the coordinates of \( G \)

\[
x_G = \frac{(1 \cdot 6) + (3 \cdot 1)}{1 + 3} = \frac{6 + 3}{4} = \frac{9}{4} = 2.25
\]

\[
y_G = \frac{(1 \cdot 4) + (3 \cdot -8)}{1 + 3} = \frac{4 - 24}{4} = \frac{-20}{4} = -5
\]

So, the coordinates of point \( G \) are \( \left(2.25, -5\right) \).

### Step 4: Plot the point \( G \)
On the graph, locate the point \( G \) at \( (2.25, -5) \). This point should lie on the line segment \( EF \).

Would you like further details, or have any questions on this solution? 

Here are some related questions you might explore:
1. How does the section formula apply when \( G \) divides \( EF \) externally?
2. What would be the coordinates of \( G \) if the ratio \( EG:FG \) was \( 2:3 \)?
3. How can the midpoint formula be derived from the section formula?
4. If point \( G \) divided \( EF \) in a different ratio, how would you calculate it?
5. How does the concept of weighted averages relate to the section formula?

**Tip:** Understanding the section formula can be particularly useful when solving problems involving line segments and ratios, especially in coordinate geometry.

Point G lies on EF such that EG:FG is 1:3. Graph G.

Learn how to find the coordinates of point G on line segment EF, where EG:FG is 1:3. This problem uses the section formula and is ideal for students in Grades 9-10 studying coordinate geometry and ratios.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation