Lesson 4: Polygons in the Coordinate Plane

Property

A geometric shape can be drawn on a coordinate plane by plotting its vertices (corners) as ordered pairs $(x, y)$ and connecting them with line segments.

Examples

Property

The distance $d$ between two points on a horizontal or vertical line is found using their differing coordinates, $a$ and $b$:

• If the points are on opposite sides of an axis, add their absolute values: $d = |a| + |b|$.
• If the points are on the same side of an axis, subtract the smaller absolute value from the larger: $d = |\text{larger absolute value}| - |\text{smaller absolute value}|$.

Examples

Property

The perimeter of a rectangle is the sum of its four side lengths. For a rectangle with horizontal and vertical sides, the lengths can be found using the coordinates of its vertices. The perimeter $P$ is calculated by adding the lengths of the two horizontal sides and the two vertical sides.

Examples

• A rectangle has vertices at $(-2, 3)$, $(4, 3)$, $(4, -1)$, and $(-2, -1)$. The horizontal sides have length $|4 - (-2)| = 6$ units. The vertical sides have length $|3 - (-1)| = 4$ units. The perimeter is $P = 6 + 6 + 4 + 4 = 20$ units.
• A square has vertices at $(1, 1)$, $(5, 1)$, $(5, 5)$, and $(1, 5)$. All sides have length $|5 - 1| = 4$ units. The perimeter is $P = 4 + 4 + 4 + 4 = 16$ units.

Explanation

To find the perimeter of a rectangle on a coordinate plane, first determine the lengths of its horizontal and vertical sides. Use the absolute value of the difference in the x-coordinates for horizontal lengths and the y-coordinates for vertical lengths. Since a rectangle has two pairs of equal-length sides, you will calculate one horizontal length and one vertical length. Finally, add all four side lengths together to find the total perimeter of the rectangle.

Property

The perimeter, $P$, of a polygon on the coordinate plane is the sum of the lengths of all its sides. For a polygon with $n$ sides of lengths $s_1, s_2, ..., s_n$, the perimeter is calculated as:

The lengths of horizontal and vertical sides are found by calculating the distances between their vertices.

Examples

Find the perimeter of a polygon with vertices $A(-2, 1)$, $B(3, 1)$, $C(3, -3)$, and $D(-2, -3)$.

• The side lengths are $AB = |3 - (-2)| = 5$, $BC = |1 - (-3)| = 4$, $CD = |3 - (-2)| = 5$, and $DA = |-3 - 1| = 4$.
• The perimeter is $P = 5 + 4 + 5 + 4 = 18$ units.

Find the perimeter of a polygon with vertices $P(1, 4)$, $Q(5, 4)$, $R(5, 2)$, $S(3, 2)$, $T(3, 1)$, and $U(1, 1)$.

• The side lengths are $PQ=4$, $QR=2$, $RS=2$, $ST=1$, $TU=2$, $UP=3$.
• The perimeter is $P = 4 + 2 + 2 + 1 + 2 + 3 = 14$ units.