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Find side lengths of coordinate-plane polygons

Side lengths of coordinate-plane polygons are determined by comparing the coordinates of consecutive vertices: for a horizontal side, take the absolute difference of the xx-coordinates, and for a vertical side, take the absolute difference of the yy-coordinates. The length represents the number of coordinate units between endpoints, including cases involving negative coordinates; general distance formulas for diagonal sides and irrational lengths are outside this understanding.

Detailed Explanation: 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 yy-coordinates are the same. Subtract the xx-coordinates and use the absolute value:
length=∣x2−x1∣\text{length}= \vert x_2-x_1 \vert
  • For a vertical side, the xx-coordinates are the same. Subtract the yy-coordinates and use the absolute value:
length=∣y2−y1∣\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 ABCDABCD with vertices

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

Compare each pair of consecutive vertices:

  1. Side AB‾\overline{AB}

    The yy-coordinates are both 22, so this side is horizontal.

∣3−(−4)∣=∣7∣=7 \vert 3-(-4) \vert = \vert 7 \vert =7

So, AB‾\overline{AB} is 77 units long.

  1. Side BC‾\overline{BC}

    The xx-coordinates are both 33, so this side is vertical.

∣−1−2∣=∣−3∣=3 \vert -1-2 \vert = \vert -3 \vert =3

So, BC‾\overline{BC} is 33 units long.

  1. Side CD‾\overline{CD}

    The yy-coordinates are both −1-1, so this side is horizontal.

∣−4−3∣=∣−7∣=7 \vert -4-3 \vert = \vert -7 \vert =7

So, CD‾\overline{CD} is 77 units long.

  1. Side DA‾\overline{DA}

    The xx-coordinates are both −4-4, so this side is vertical.

∣2−(−1)∣=∣3∣=3 \vert 2-(-1) \vert = \vert 3 \vert =3

So, DA‾\overline{DA} is 33 units long.

Therefore, the side lengths are

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

Learn by doing: Find side lengths of coordinate-plane polygons

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    Cartesian Grid - Diagram Coordinates to Line Length - Including Negative

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