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Given a parallelogram ABCD
Vertex B is on the X-axis and the diagonal DB is perpendicular to side AB
given A(0, -10)

Let's break down the problem from the image step by step to solve it. The information we have is:

- The shape is a parallelogram \(ABCD\).
- Vertex \(A(0, -10)\) is given.
- Vertex \(B\) lies on the x-axis.
- The diagonal \(DB\) is perpendicular to side \(AB\).
- The equation of line \(DC\) is provided as \(y = \frac{2}{5}x + 19\).

### Step-by-Step Solution:

1. **Find the coordinates of vertex B:**
   Since \(B\) lies on the x-axis, its \(y\)-coordinate must be 0. We will denote the coordinates of \(B\) as \(B(x_B, 0)\).

2. **Find the slope of line AB:**
   The slope of line \(AB\), denoted as \(m_{AB}\), can be calculated by the formula:
   \[
   m_{AB} = \frac{y_B - y_A}{x_B - x_A}
   \]
   Since \(A(0, -10)\) and \(B(x_B, 0)\), the slope becomes:
   \[
   m_{AB} = \frac{0 - (-10)}{x_B - 0} = \frac{10}{x_B}
   \]

3. **Condition for perpendicularity of DB and AB:**
   Since \(DB\) is perpendicular to \(AB\), the product of their slopes must satisfy:
   \[
   m_{AB} \cdot m_{DB} = -1
   \]
   The slope of \(DB\) can be written as:
   \[
   m_{DB} = \frac{y_B - y_D}{x_B - x_D}
   \]
   But first, let's calculate the slope of \(DB\) and determine further based on perpendicularity and the coordinates of \(B\) and \(D\).

4. **Equation of line DC:**
   We are given that the line \(DC\) has the equation:
   \[
   y = \frac{2}{5}x + 19
   \]
   We will use this equation to find the coordinates of point \(D\) or point \(C\) later in the process.

#### Proceeding with the calculations:
I will calculate the values of the coordinates for vertices \(B\) and \(D\). Let's do that now.It seems there was an issue with calculating the perpendicularity condition symbolically. I will try an alternative approach by first solving for coordinates and relations based on the constraints provided.

Let's break the problem into smaller steps and focus first on determining vertex \(B\) using the condition that diagonal \(DB\) is perpendicular to \(AB\).

To proceed, I'll compute the coordinates of vertex \(B\) considering the geometric constraints and properties of the parallelogram.The solution for the \(y\)-coordinate of point \(D\) is:

\[
y_D = \frac{x_B (x_B - x_D)}{10}
\]

This gives us a relationship between \(x_B\), \(x_D\), and \(y_D\), which we'll use to find the final coordinates of \(D\) once \(B\) and \(D\)'s relationship with line \(DC\) is fully incorporated.

Next, I will use the equation of line \(DC\) given as \(y = \frac{2}{5}x + 19\) to substitute and find more values.

Given a parallelogram ABCD, where vertex B is on the X-axis and the diagonal DB is perpendicular to side AB, and given A(0, -10), find the necessary coordinates and properties of the parallelogram.

Explore how to solve a geometry problem involving a parallelogram positioned on a coordinate plane, with specific details on vertex positions and side relationships. This analysis helps in understanding the properties of parallelograms and coordinate geometry.

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