Skill: Find side lengths of coordinate-plane polygons

To find the side lengths of a coordinate-plane polygon, compare the coordinates of consecutive vertices.

• For a horizontal side, the $y$-coordinates are the same. Subtract the $x$-coordinates and use the absolute value:

$$\text{length}= \vert x_2-x_1 \vert$$

• For a vertical side, the $x$-coordinates are the same. Subtract the $y$-coordinates and use the absolute value:

$$\text{length}= \vert y_2-y_1 \vert$$

The absolute value makes the length positive, even when coordinates are negative.

Example: Find the side lengths of polygon $ABCD$ with vertices

$$A(-4,2),\quad B(3,2),\quad C(3,-1),\quad D(-4,-1).$$

Compare each pair of consecutive vertices:

1. Side $\overline{AB}$

   The $y$-coordinates are both $2$, so this side is horizontal.

$$\vert 3-(-4) \vert = \vert 7 \vert =7$$

So, $\overline{AB}$ is $7$ units long.

2. Side $\overline{BC}$

   The $x$-coordinates are both $3$, so this side is vertical.

$$\vert -1-2 \vert = \vert -3 \vert =3$$

So, $\overline{BC}$ is $3$ units long.

3. Side $\overline{CD}$

   The $y$-coordinates are both $-1$, so this side is horizontal.

$$\vert -4-3 \vert = \vert -7 \vert =7$$

So, $\overline{CD}$ is $7$ units long.

4. Side $\overline{DA}$

   The $x$-coordinates are both $-4$, so this side is vertical.

$$\vert 2-(-1) \vert = \vert 3 \vert =3$$

So, $\overline{DA}$ is $3$ units long.

Therefore, the side lengths are

$$\boxed{7\text{ units},\ 3\text{ units},\ 7\text{ units},\ 3\text{ units}}.$$