https://file.aiquickdraw.com/mathbot/file/1725595519045x9wjzqcx.jpg

To solve this problem, we need to find the point \( Y \) on line segment \( CD \) such that the ratio of \( DY \) to \( YC \) is \( 2:1 \).

### Step-by-step breakdown:

1. **Identify the coordinates of points \( C \) and \( D \):**
   - From the grid, we see that \( C \) is at \( (2, 3) \) and \( D \) is at \( (7, -1) \).

2. **Find the coordinates of point \( Y \):**
   - We know that the ratio \( \frac{DY}{YC} = \frac{2}{1} \). This means point \( Y \) divides the line segment \( CD \) in a ratio of \( 2:1 \), with point \( D \) closer to \( Y \) than \( C \).
   - The formula for the coordinates of a point dividing a line segment in a given ratio is given by:
     \[
     Y = \left( \frac{m \cdot x_2 + n \cdot x_1}{m+n}, \frac{m \cdot y_2 + n \cdot y_1}{m+n} \right)
     \]
     where \( (x_1, y_1) \) are the coordinates of point \( C \), \( (x_2, y_2) \) are the coordinates of point \( D \), and the ratio is \( m:n \).

3. **Substitute the values:**
   - \( C = (2, 3) \) and \( D = (7, -1) \)
   - The ratio is \( m:n = 2:1 \)
   
   Substituting into the formula:
   \[
   Y_x = \frac{2 \cdot 7 + 1 \cdot 2}{2 + 1} = \frac{14 + 2}{3} = \frac{16}{3} \approx 5.33
   \]
   \[
   Y_y = \frac{2 \cdot (-1) + 1 \cdot 3}{2 + 1} = \frac{-2 + 3}{3} = \frac{1}{3} \approx 0.33
   \]

4. **Conclusion:**
   - The coordinates of point \( Y \) are approximately \( (5.33, 0.33) \).
   
Thus, point \( Y \) divides the segment \( CD \) in the ratio \( 2:1 \), with \( DY: YC = 2:1 \).

Would you like further clarification or additional details?

### Related Questions:
1. How can we apply the section formula for different ratios?
2. What happens if the ratio was reversed, \( 1:2 \), instead of \( 2:1 \)?
3. How can we find the midpoint of a line segment between two points?
4. What is the distance between points \( C \) and \( D \) in this problem?
5. How do you interpret the slope of a line from the grid?

### Tip:
Always check the scale of the grid when reading coordinates to ensure accuracy, especially when points are located between grid lines!

Refer to the coordinate grid. Find point Y on CD such that the ratio of DY to YC is 2:1.

Explore how to find point Y on line segment CD with a 2:1 ratio division using coordinate geometry and the section formula. This guide provides step-by-step instructions suitable for high school students in grades 9-12.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation