To solve the problem, let's analyze each part step-by-step.
Given Data:
- Point \( E(-3, 2) \) is the intersection of lines \( AB \) and \( DC \).
- Point \( B \) is on the \( x \)-axis, so its coordinates are \( (b, 0) \).
- Line \( DC \) intersects the \( y \)-axis at \( -1 \), thus the \( y \)-intercept of line \( DC \) is \( (0, -1) \).
- \( \angle XFE = 135^\circ \), where \( F \) is the \( x \)-intercept of \( AB \).
- The equation of line \( AD \) is given as \( y - 4 = 0 \), i.e., \( y = 4 \).
- Lines \( DC \) and \( AB \) are perpendicular.
Step 4.1: The Gradient of Line \( DC \)
To find the gradient of line \( DC \), we can use the coordinates of the point where it intersects the \( y \)-axis:
- The coordinates are \( (0, -1) \) (y-intercept) and point \( E(-3, 2) \).
The gradient (\( m \)) of a line through points \( (x_1, y_1) \) and \( (x_2, y_2) \) is given by the formula:
\[ m = \frac{y_2 - y_1}{x_2 - x_1} \]
Substituting the points into the formula:
\[ m_{DC} = \frac{2 - (-1)}{-3 - 0} = \frac{2 + 1}{-3} = \frac{3}{-3} = -1 \]
So, the gradient of \( DC \) is \( -1 \) (Answer: -1).
Step 4.2: The Equation of Line \( DC \)
We can use the gradient-intercept form of the equation of a line, which is \( y = mx + c \), where \( m \) is the gradient and \( c \) is the \( y \)-intercept.
Here, since \( m = -1 \) and \( c = -1 \):
\[ y = -1x - 1 \quad \text{or} \quad y = -x - 1 \]
So, the equation of line \( DC \) is \( y = -x - 1 \) (Answer: y = -x - 1).
Step 4.3: The Coordinates of Point \( D \)
To find the coordinates of point \( D \), we know:
- Line \( AD \) is horizontal (since \( y = 4 \)).
- Point \( D \) lies on \( y = 4 \) and also lies on line \( DC \).
To find the intersection of line \( DC: y = -x - 1 \) and line \( AD: y = 4 \):
Set \( -x - 1 = 4 \):
\[ -x - 1 = 4 \implies -x = 5 \implies x = -5 \]
Thus, the coordinates of point \( D \) are \( (-5, 4) \) (Answer: (-5, 4)).
Step 4.4: Size of Angle \( DAB \)
Since \( DA \parallel BF \), and \( DC \perp AB \), we can find the angle \( DAB \):
- The gradient of line \( AB \) is \( m_{AB} = 1 \) (since \( DC \) has a gradient of \( -1 \))
- We can use the tangent of angles to find angle \( DAB \):
We can find the angles using the gradient relationship:
\[ \tan(\theta) = \frac{m_{AB} - m_{BC}}{1 + m_{AB} \cdot m_{BC}} \]
Using our gradients:
- Gradient \( m_{AB} = 1 \)
- Gradient \( m_{DC} = -1 \)
To find \( \angle DAB \):
The angle \( DAB \) is complementary to \( XFE \), given \( XFE = 135^\circ \).
Thus:
\[ \theta_{DAB} = 90^\circ - 135^\circ = 45^\circ \]
So, the size of angle \( DAB \) is \( 45^\circ \) (Answer: 45Β°).
Summary of Answers:
4.1: -1
4.2: y = -x - 1
4.3: (-5, 4)
4.4: 45Β°