https://file.aiquickdraw.com/mathbot/file/17255627910320ufu8fjc.jpg

The problem asks to find the coordinates of point \( E \) that partitions the line segment \( DF \) in a given ratio. Here's the information provided:

- Point \( D(-1, 11) \)
- Point \( F(9, -5) \)
- Ratio of \( DE \) to \( DF \) is \( 5:8 \)

### Step 1: Formula for finding the coordinates of a point partitioning a line segment
We use the section formula to find the coordinates of point \( E(x_E, y_E) \). If a point divides a line segment in the ratio \( m:n \), the coordinates of the point are given by:
\[
x_E = \frac{m \cdot x_2 + n \cdot x_1}{m + n}
\]
\[
y_E = \frac{m \cdot y_2 + n \cdot y_1}{m + n}
\]
where \( (x_1, y_1) \) and \( (x_2, y_2) \) are the coordinates of the two endpoints, and \( m:n \) is the ratio.

Here, \( D(-1, 11) \) corresponds to \( (x_1, y_1) \), \( F(9, -5) \) corresponds to \( (x_2, y_2) \), and the ratio \( DE:DF = 5:8 \) implies \( m = 5 \) and \( n = 3 \) (since \( DF \) is split as \( DE = 5 \) and the remaining \( EF = 8 - 5 = 3 \)).

### Step 2: Calculate the coordinates of \( E \)
Now, I'll calculate the coordinates of \( E \) using the section formula.The coordinates of point \( E \), which partitions the line segment \( DF \) in the ratio \( 5:8 \), are \( (5.25, 1.0) \).

Would you like more details on any part of this solution?

Here are five related questions to consider:
1. How is the section formula derived, and when is it useful?
2. What is the difference between internal and external division of a line segment?
3. How would the calculation change if the ratio were reversed (e.g., \( DE:DF = 3:5 \))?
4. How can this method be applied in three-dimensional geometry?
5. What happens if the ratio between \( DE \) and \( DF \) is not a whole number?

**Tip**: Always check the total ratio when partitioning a line segment. If the ratio sums to the total length, you're partitioning internally; otherwise, it's an external partition.

Given DF with D(-1, 11) and F(9, -5), if E partitions DF such that the ratio of DE to DF is 5:8, find the coordinates of E.

Learn how to find the coordinates of a point that partitions a line segment DF with points D(-1, 11) and F(9, -5) in the ratio of 5:8. This problem uses the section formula and involves fundamental concepts of coordinate geometry and ratios. Ideal for students in Grades 9-11.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation