Lesson 3: Classify Quadrilaterals

Property

A quadrilateral can be classified by the properties of its sides. On a coordinate plane, we can use coordinates to determine if sides are horizontal or vertical.

• Horizontal Line: Two points have the same y-coordinate, like $(x_1, y)$ and $(x_2, y)$.
• Vertical Line: Two points have the same x-coordinate, like $(x, y_1)$ and $(x, y_2)$.
• A rectangle has four right angles, which can be formed by horizontal and vertical sides.

Examples

• The points $A(2, 1)$, $B(7, 1)$, $C(7, 5)$, and $D(2, 5)$ form a rectangle. Sides $\overline{AB}$ and $\overline{DC}$ are horizontal, and sides $\overline{AD}$ and $\overline{BC}$ are vertical.
• The points $P(-1, -1)$, $Q(3, -1)$, $R(3, 3)$, and $S(-1, 3)$ form a square, which is a special type of rectangle.
• The points $W(1, 1)$, $X(6, 1)$, $Y(8, 4)$, and $Z(3, 4)$ form a parallelogram. The sides $\overline{WX}$ and $\overline{ZY}$ are parallel because they are both horizontal.

Explanation

By plotting the vertices of a quadrilateral on a coordinate plane, we can classify its shape. Check the coordinates of the vertices to identify horizontal and vertical sides. If a quadrilateral has two pairs of opposite sides that are horizontal and vertical, it must be a rectangle. This method uses the properties of lines on the coordinate plane to understand geometric figures.

Property

There are two common definitions for a trapezoid:

• Inclusive Definition: A quadrilateral with at least one pair of parallel sides.
• Exclusive Definition: A quadrilateral with exactly one pair of parallel sides.

Examples

• A quadrilateral with vertices at $A(1, 1)$, $B(7, 1)$, $C(5, 4)$, and $D(2, 4)$ is a trapezoid under both definitions because it has exactly one pair of parallel sides ($AB$ and $CD$).
• A parallelogram with vertices at $P(1, 1)$, $Q(6, 1)$, $R(7, 4)$, and $S(2, 4)$ has two pairs of parallel sides. It is considered a trapezoid only under the inclusive definition.

Explanation

The definition of a trapezoid can vary depending on the curriculum. The inclusive definition, stating a trapezoid has at least one pair of parallel sides, is more common in higher-level mathematics. This means that under the inclusive rule, all parallelograms, rectangles, rhombuses, and squares are also considered trapezoids. The exclusive definition requires exactly one pair of parallel sides, which excludes parallelograms.