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

Given that point \( C(6,9) \) is located on the segment between points \( A(4,8) \) and \( B(x,y) \), and point \( C \) is \(\frac{1}{4}\) of the distance from \( A \) to \( B \), we can use the section formula to find the coordinates of \( B \).

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

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

Here, \( (x_1, y_1) \) are the coordinates of \( A \), \( (x_2, y_2) \) are the coordinates of \( B \), and \( (x_C, y_C) \) are the coordinates of \( C \). Given that point \( C \) is \( \frac{1}{4} \) of the distance from \( A \) to \( B \), the ratio \( m:n \) is \( 1:3 \).

Plugging in the given values:

\[
(6, 9) = \left( \frac{3x + 4 \times 4}{3+1}, \frac{3y + 4 \times 8}{3+1} \right)
\]

Simplifying this:

\[
6 = \frac{3x + 16}{4}
\]
\[
9 = \frac{3y + 32}{4}
\]

Now, solve for \( x \) and \( y \):

### Solve for \( x \):
\[
6 \times 4 = 3x + 16
\]
\[
24 = 3x + 16
\]
\[
3x = 8
\]
\[
x = \frac{8}{3}
\]

### Solve for \( y \):
\[
9 \times 4 = 3y + 32
\]
\[
36 = 3y + 32
\]
\[
3y = 4
\]
\[
y = \frac{4}{3}
\]

So, the coordinates of point \( B \) are \( \left(\frac{8}{3}, \frac{4}{3}\right) \).

Would you like more details or have any questions?

Here are five related questions you might find useful:
1. How can the section formula be applied to find a point dividing a line segment externally?
2. What is the midpoint formula, and how does it relate to the section formula?
3. Can you find the length of segment \( AB \) using the distance formula?
4. How would the coordinates of point \( B \) change if \( C \) were \( \frac{1}{3} \) of the distance from \( A \) to \( B \)?
5. How do you find the coordinates of a point that divides a line segment into a given ratio in 3D space?

**Tip:** When solving problems involving the section formula, always double-check whether the point is dividing the segment internally or externally, as it affects the ratios used in calculations.

Learn how to find the coordinates of point B when point C is located on the segment between points A and B, and C is 1/4 of the distance from A to B. Detailed step-by-step solution using the section formula in coordinate geometry.

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