https://file.aiquickdraw.com/mathbot/file/1728147844048vewgv19k.jpg

The problem asks to find the coordinates of points \(C\), \(D\), and \(B\), given the coordinates of point \(A\), and some ratio conditions for line segments. Let's break it down:

### Given:
- Coordinates of \( A(-2, 6) \).
- Line segment \( AB \) is intercepted by the \( y \)-axis at \( C \) and the \( x \)-axis at \( D \).
- The ratio \( AC : AD = 1 : 4 \), meaning \( \frac{AC}{AD} = \frac{1}{4} \).
- \( D \) is the midpoint of \( CB \).

We can solve this step by step using the section formula and midpoint formula:

### Step 1: Find coordinates of point \( D \)
Since \( D \) lies on the \( x \)-axis, its coordinates will be \( D(x_1, 0) \). We use the section formula in the ratio \( 1 : 4 \) to find the coordinates of \( D \).

The section formula for dividing a line segment joining \( (x_1, y_1) \) and \( (x_2, y_2) \) in the ratio \( m:n \) is:

\[
(x, y) = \left( \frac{m x_2 + n x_1}{m + n}, \frac{m y_2 + n y_1}{m + n} \right)
\]

Here, point \( C(0, y_C) \) lies on the \( y \)-axis (as it is the y-intercept). Applying the section formula to divide the segment \( AC \):

\[
x_D = \frac{1 \cdot x_D + 4 \cdot (-2)}{1 + 4}
= \frac{-8}{5}


In the given figure, if the line segment AB is intercepted by the y-axis and x-axis at C and D, respectively, such that AC:AD = 1:4 and D is the midpoint of CB, find the coordinates of D, C, and B.

Learn how to find the coordinates of points D, C, and B on an intercepted line using section and midpoint formulas. Given AC:AD = 1:4 and D is the midpoint of CB, apply key concepts of coordinate geometry.

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