 The coordinates of points A,B,C,D are respectively (3,1),(1,0),(5,1),(-10,-4). Find the ratio in which CD divides AB

To find the ratio in which line \( CD \) divides line \( AB \), we can use the section formula. The section formula helps determine the ratio in which a point divides a line segment.

Let's consider points \( A(3, 1) \), \( B(1, 0) \), \( C(5, 1) \), and \( D(-10, -4) \). We want to find the ratio \( k \) in which the line joining \( CD \) divides \( AB \).

### Step 1: Apply the section formula
The section formula states that if a point \( P(x, y) \) divides a line segment joining two points \( (x_1, y_1) \) and \( (x_2, y_2) \) in the ratio \( m:n \), then the coordinates of the point \( P \) are given by:

\[
P(x, y) = \left( \frac{mx_2 + nx_1}{m+n}, \frac{my_2 + ny_1}{m+n} \right)
\]

Here, \( P \) could either be \( C \) or \( D \). Let’s first assume that point \( C \) divides \( AB \) in some ratio \( k:1 \).

### Step 2: Apply coordinates of point \( C(5, 1) \)
Using the section formula for the coordinates of point \( C(5, 1) \) that divides \( A(3, 1) \) and \( B(1, 0) \) in the ratio \( k:1 \):

\[
5 = \frac{k \times 1 + 1 \times 3}{k+1} \quad \text{(for the x-coordinate)}
\]
\[
1 = \frac{k \times 0 + 1 \times 1}{k+1} \quad \text{(for the y-coordinate)}
\]

### Step 3: Solve for \( k \)
Start with the y-coordinate equation:
\[
1 = \frac{1}{k+1}
\]

Multiplying both sides by \( k+1 \):
\[
k+1 = 1 \quad \Rightarrow \quad k = 0
\]

Thus, point \( C \) divides the line segment \( AB \) in the ratio \( 0:1 \), meaning \( C \) coincides with point \( A \).

### Step 4: Check for point \( D(-10, -4) \)
Using the same approach, check if point \( D(-10, -4) \) divides \( AB \) in a different ratio. However, based on point \( C \) coinciding with \( A \), the problem might involve more complex geometric interpretations.

Would you like to proceed exploring this or adjust the problem approach? Here are 5 related problems:
1. How to find the equation of line passing through points \( A \) and \( B \)?
2. How to calculate the area of a triangle using coordinates of three points?
3. What is the distance between two points in a coordinate plane?
4. How to find the midpoint of a line segment joining two points?
5. What is the slope of the line joining points \( A(3,1) \) and \( B(1,0) \)?

**Tip:** Always check the alignment of points when dealing with ratios and line segments.

The coordinates of points A,B,C,D are respectively (3,1),(1,0),(5,1),(-10,-4). Find the ratio in which CD divides AB.

Learn how to find the ratio in which the line CD divides the line AB using coordinate geometry and the section formula. Given the coordinates of points A(3,1), B(1,0), C(5,1), and D(-10,-4), this detailed explanation is perfect for students in Grades 9-11.

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