(4) On a pair of axes, construct a rhombus. Find the equation of each side. What do you notice? Investigate for different quadrilaterals.

To construct a rhombus on a pair of axes, we define its vertices. Let's choose the vertices $A=(0,3)$, $B=(4,0)$, $C=(0,-3)$, and $D=(-4,0)$.

Using points $A(0,3)$ and $B(4,0)$, the slope is $m_{AB} = \frac{0-3}{4-0} = -\frac{3}{4}$.
The equation of the line is $y - 3 = -\frac{3}{4}(x - 0)$.

Using points $B(4,0)$ and $C(0,-3)$, the slope is $m_{BC} = \frac{-3-0}{0-4} = \frac{-3}{-4} = \frac{3}{4}$.
The equation of the line is $y - 0 = \frac{3}{4}(x - 4)$.

Using points $C(0,-3)$ and $D(-4,0)$, the slope is $m_{CD} = \frac{0-(-3)}{-4-0} = \frac{3}{-4} = -\frac{3}{4}$.
The equation of the line is $y - (-3) = -\frac{3}{4}(x - 0)$.

Using points $D(-4,0)$ and $A(0,3)$, the slope is $m_{DA} = \frac{3-0}{0-(-4)} = \frac{3}{4}$.
The equation of the line is $y - 0 = \frac{3}{4}(x - (-4))$.

We notice that:

• Opposite sides have the same slope ($m_{AB} = m_{CD} = -\frac{3}{4}$ and $m_{BC} = m_{DA} = \frac{3}{4}$), indicating that opposite sides are parallel.
• The equations of opposite sides (e.g., $3x+4y=12$ and $3x+4y=-12$) share the same $x$ and $y$ coefficients, differing only in the constant term. This is characteristic of parallel lines.

• A parallelogram will also have opposite sides with the same slope, meaning their equations will differ only by a constant.
• A rectangle is a parallelogram with adjacent sides perpendicular. This means the product of the slopes of adjacent sides will be $-1$.
• A square is a rectangle with all sides equal. It will exhibit both parallel opposite sides and perpendicular adjacent sides.
• A trapezium (or trapezoid) has only one pair of opposite sides that are parallel, so only two of its side equations will have the same slope.
• A kite has two pairs of equal-length adjacent sides. There is no general rule for the slopes of its sides, but its diagonals are perpendicular.
• A general quadrilateral has no specific relationships between the slopes or equations of its sides.