Answer the following. Are the points A(1,1), B(5,2),...

$$\text { Answer the following.}$$
Are the points $A(1,1), B(5,2), C(3,4),$ and $D(-1,3)$ the vertices of a parallelogram (opposite sides equal in length)? of a rhombus (all sides equal in length)?

Key Concepts

Properties of Rhombuses

A rhombus is a special type of parallelogram where all four sides are of equal length, and its diagonals are perpendicular bisectors of each other. Establishing the equality of all side lengths through distance calculations and examining the behavior of the diagonals are key steps in determining if a quadrilateral is a rhombus.

Midpoint Formula

The midpoint formula calculates the point that lies exactly halfway between two endpoints of a segment in the coordinate plane. It is particularly useful for verifying properties of quadrilaterals, such as whether the diagonals bisect each other—an important criterion in identifying a parallelogram.

Properties of Parallelograms

A parallelogram is defined by the property that both pairs of opposite sides are parallel and equal in length. Additionally, its diagonals bisect each other. Verifying these properties using algebraic methods in the coordinate plane is critical in confirming whether a set of points forms a parallelogram.

Distance Formula

The distance formula, derived from the Pythagorean theorem, is used to calculate the distance between any two points in the coordinate plane. This tool is essential for comparing side lengths and confirming whether segments are equal in length, a key step in determining if a quadrilateral is a parallelogram or a rhombus.

Coordinate Geometry

Coordinate geometry provides a framework for analyzing geometric figures by placing them in the coordinate plane, enabling the use of algebraic formulas to compute distances, slopes, and midpoints. This approach allows for a rigorous verification of geometrical properties such as parallelism, equality of lengths, and bisecting lines.

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