# Geometry of the Plane

## Worked Examples

## Quadrilaterals

There are several kinds of quadrilaterals. We can identify them by their vertices and slopes.**Rectangles**are quadrilaterals containing ony right angles. If the sides have equal lengths, then it is called a

**square**.

We can check that we have a rectangle by checking the slopes of the sides: they are parallel or perpendicular.

**Parallelogram**are quadrilaterals with opposite sides parallel, but adjacent sides not perpendicular. If the sides have equal lengths, then it is called a

**rhombus**.

**Trapezoid**are quadrilaterals where

**one**pair of opposite sides are parallel

### Example.

Move the points $A$, $B$, $C$, and $D$ to obtain different quadrilaterals.### Example.

What kind of quadrilateral has the following vertices: $(0,5)$, $(1,2)$, $(4,8)$, and $(5,5)$.**Solution.**First we draw these four points:

We now check the slopes of the sides of the quadrilateral: \begin{align} \text{slope of } AB & = \frac{2-5}{1-0} = -3 \\ \text{slope of } BD & = \frac{5-2}{5-1} = \frac34 \\ \text{slope of } AC & = \frac{8-5}{4-0} = \frac34 \\ \text{slope of } CD & = \frac{5-8}{5-4} = -3 \end{align} Observe that adjacent sides ($AB$ and $BD$) are not perpendicular (Details)Try to show why they are not perpendicular, so it is not a rectangle or a square.

Also, we have two pairs of parallel sides: $AB$ is parallel to $CD$, and $BD$ is parallel to $AC$.

So it is a parallelogram or a rhombus.

To distinguish between them, we need to check the length of the sides: \begin{align} {\rm distance}(A,B) & = \sqrt{(2-5)^2+(1-0)^2} = \sqrt{10} \\ {\rm distance}(B,D) & = \sqrt{(5-2)^2+(5-1)^2} = 5 \end{align} (Details)The distance formula will be reviewed in the next page (Other Formulas)

Since the length of the sides are not equal, the quadrilateral is a

**parallelogram**.