Let's break down the given problem using the information provided:

We are given a rectangle ABCD.

Coordinates:

1. $A(0,0)$ lies on the line $y = 2x$.
2. $C(3, 0)$ lies on the line $y = 2x - 6$.

From the image, the rectangle has one side along the x-axis (line AC), and lines AB and CD are parallel and perpendicular to this axis.

We are asked for:

1. The equation of the line $AD$.
2. The coordinates of point $D$.
3. The area of the rectangle $ABCD$.

1. Equation of Line AD:

Since AD is a vertical line (parallel to the y-axis), its equation will be of the form $x = k$, where $k$ is the x-coordinate of points $A$ and $D$. So, the equation of line AD is $x = 0$, as point $A(0,0)$.

2. Coordinates of Point D:

To find the coordinates of point D, we need to recognize that since AD is vertical, the x-coordinate of point D will be 0. The y-coordinate will depend on how far point D extends from point C since AD is perpendicular to AC and lies along the vertical.

We'll compute this explicitly by recognizing that the slope of line CD is the same as line AB because AB and CD are parallel.

3. Area of Rectangle ABCD:

The area of the rectangle is determined by the length of the sides. One side is the distance between points A and D (vertical length), and the other is the distance between points A and C (horizontal length).

I will compute these values explicitly now to provide the correct answers.import sympy as sp

Given points and equations

x, y = sp.symbols('x y')

Line equations y = 2x and y = 2x - 6

line_AB = 2x line_CD = 2x - 6

Points A and C

A = (0, 0) C = (3, 0)

Equation of line AD is x = 0 (vertical line)

Now, we solve for the coordinates of point D

Line AD is vertical, so x-coordinate of D is the same as A's x-coordinate (0)

We substitute x = 0 in the equation of line CD (to find D's y-coordinate)

y_coordinate_D = line_CD.subs(x, 0)

Calculate the area of the rectangle ABCD

Horizontal length is AC = |x_C - x_A| = 3 - 0 = 3 units

Vertical length is AD = |y_D - y_A| = |y_coordinate_D - 0|

AC_length = C[0] - A[0] # Horizontal length AD_length = abs(y_coordinate_D - A[1]) # Vertical length

Area of the rectangle is the product of AC and AD lengths

area_ABCD = AC_length * AD_length

y_coordinate_D, (0, y_coordinate_D), area_ABCDHere are the results for the given rectangle problem:

1. Equation of the line AD: Since it's a vertical line, the equation is $x = 0$.
2. Coordinates of point D: $D(0, -6)$.
3. Area of the rectangle ABCD: The area is 18 square units.

Let me know if you'd like further clarification or if you have any other questions.

Here are 5 related questions you can consider:

1. What would the area be if the slope of line AB changed?
2. How would the coordinates of point D change if the rectangle were tilted?
3. What is the relationship between the slopes of lines AB and CD?
4. How would the area change if the line AC were shifted vertically?
5. What is the formula for the distance between two points in general?

Tip: When working with perpendicular lines in geometry, remember that the product of their slopes is always $-1$.