point c (6,9) is located on the segment between point a(4,8) and point b. c is 1/4 the distance from a to b. what are the coordinates of point B?

To find the coordinates of point \( B \), we can use the section formula. Since point \( C(6, 9) \) divides the segment \( AB \) in the ratio \( 1:3 \), we can set up the section formula accordingly.

Let the coordinates of point \( B \) be \( (x_2, y_2) \).

The section formula for a point dividing a line segment internally in the ratio \( m:n \) is given by:

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

Given:
- \( m = 1 \), \( n = 3 \)
- \( A(4, 8) = (x_1, y_1) \)
- \( C(6, 9) = (x, y) \)

Substitute the values into the section formula:

\[
6 = \frac{1 \cdot x_2 + 3 \cdot 4}{1 + 3} = \frac{x_2 + 12}{4}
\]

\[
9 = \frac{1 \cdot y_2 + 3 \cdot 8}{1 + 3} = \frac{y_2 + 24}{4}
\]

Now, solve these equations for \( x_2 \) and \( y_2 \):

1. For the \( x \)-coordinate:

\[
6 \times 4 = x_2 + 12
\]
\[
24 = x_2 + 12
\]
\[
x_2 = 24 - 12 = 12
\]

2. For the \( y \)-coordinate:

\[
9 \times 4 = y_2 + 24
\]
\[
36 = y_2 + 24
\]
\[
y_2 = 36 - 24 = 12
\]

Thus, the coordinates of point \( B \) are \( (12, 12) \).

Would you like details on any part of this solution, or do you have any other questions?

### Relative Questions:
1. How does the section formula work for dividing a line segment externally?
2. What would be the coordinates of point \( C \) if it was \( \frac{1}{3} \) the distance from \( A \) to \( B \)?
3. How would the answer change if \( C \) was \( 3/4 \) the distance from \( A \) to \( B \)?
4. Can you apply the section formula to three-dimensional coordinates?
5. How can we prove that the section formula is derived from the concept of similar triangles?

### Tip:
Remember that the section formula can be used both for internal and external division of a line segment by a point.

Learn how to find the coordinates of point B on a line segment using the section formula for internal division. This problem involves coordinate geometry and applies the section formula with step-by-step calculations. Suitable for Grades 9-10 students studying coordinate geometry.

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