Find the area of a rhombus if its vertices are A(3, 0), B(4, 5), C(-1, 4) and D(-2, -1). Also find the distance between its sides AB and CD.

Here are the solutions for questions 35 and 36.

Question 35: Rhombus

a) Find the area of a rhombus if its vertices are A(3, 0), B(4, 5), C(-1, 4) and D(-2, -1).

Step 1: Calculate the length of the diagonals.
The diagonals of the rhombus are AC and BD.
Length of diagonal AC:

$$AC = \sqrt{(-1-3)^2 + (4-0)^2}$$ $$AC = \sqrt{(-4)^2 + (4)^2}$$ $$AC = \sqrt{16 + 16}$$ $$AC = \sqrt{32} = 4\sqrt{2} \text{ units}$$

Length of diagonal BD:

$$BD = \sqrt{(-2-4)^2 + (-1-5)^2}$$ $$BD = \sqrt{(-6)^2 + (-6)^2}$$ $$BD = \sqrt{36 + 36}$$ $$BD = \sqrt{72} = 6\sqrt{2} \text{ units}$$

Step 2: Calculate the area of the rhombus.
The area of a rhombus is given by the formula $\frac{1}{2} d_1 d_2$, where $d_1$ and $d_2$ are the lengths of the diagonals.

$$\text{Area} = \frac{1}{2} \times AC \times BD$$ $$\text{Area} = \frac{1}{2} \times (4\sqrt{2}) \times (6\sqrt{2})$$ $$\text{Area} = \frac{1}{2} \times 24 \times (\sqrt{2})^2$$ $$\text{Area} = \frac{1}{2} \times 24 \times 2$$ $$\text{Area} = 24 \text{ square units}$$

b) Also find the distance between its sides AB and CD.

Step 1: Find the equation of line AB.
Points A(3, 0) and B(4, 5).
Slope $m_{AB} = \frac{5-0}{4-3} = \frac{5}{1} = 5$.
Using the point-slope form $y - y_1 = m(x - x_1)$:

$$y - 0 = 5(x - 3)$$ $$y = 5x - 15$$ $$5x - y - 15 = 0$$

Step 2: Find the equation of line CD.
Points C(-1, 4) and D(-2, -1).
Slope $m_{CD} = \frac{-1-4}{-2-(-1)} = \frac{-5}{-1} = 5$.
Using the point-slope form $y - y_1 = m(x - x_1)$:

$$y - 4 = 5(x - (-1))$$ $$y - 4 = 5(x + 1)$$ $$y - 4 = 5x + 5$$ $$5x - y + 9 = 0$$

Step 3: Calculate the distance between the parallel lines AB and CD.
The distance between two parallel lines $Ax + By + C_1 = 0$ and $Ax + By + C_2 = 0$ is given by $d = \frac{|C_1 - C_2|}{\sqrt{A^2 + B^2}}$.
For line AB: $5x - y - 15 = 0$ ($A=5, B=-1, C_1=-15$).
For line CD: $5x - y + 9 = 0$ ($A=5, B=-1, C_2=9$).

$$d = \frac{|-15 - 9|}{\sqrt{5^2 + (-1)^2}}$$ $$$$