**Introduction**

A four-sided geometric figure that lies in a single plane is called a *quadrilateral* . There are several classifications and various formulas that apply to each. Here are some of the more common formulas that can be useful in physics.

**Parallelogram Diagonals**

Suppose that we have a parallelogram defined by four points *P, Q, R* , and *S* . Let *D* be a line segment connecting *P* and *R* as shown in Fig. 4-22 *a* . Then *D* is a *minor diagonal* of the parallelogram, and the triangles defined by *D* are congruent:

Δ *PQR* ≅ Δ *RSP*

Let *E* be a line segment connecting Q and *S* (see Fig. 4-22 *b* ). Then *E* is a major *diagonal* of the parallelogram, and the triangles defined by *E* are congruent:

Δ *QRS* ≅ Δ *SPQ*

**Bisection Of Parallelogram Diagonals**

Suppose that we have a parallelogram defined by four points *P, Q, R* , and *S* . Let *D* be the diagonal connecting *P* and *R* ; let *E* be the diagonal connecting *Q* and *S* (Fig. 4-23). Then *D* and *E* bisect each other at their intersection point *T* . In addition, the following pairs of triangles are congruent:

Δ *PQT* ≅ Δ *RST*

Δ *QRT* ≅ Δ *SPT*

The converse of the foregoing is also true: If we have a plane quadrilateral whose diagonals bisect each other, then that quadrilateral is a parallelogram.

**Rectangle**

Suppose that we have a parallelogram defined by four points *P, Q, R* , and *S* . Suppose that any of the following statements is true for angles in degrees:

∠ *PQR* = 90° = π/2 radians

∠ *QRS* = 90° = π/2 radians

∠ *RSP* = 90° = π/2 radians

∠ *SPQ* = 90° = π/2 radians

Then all four interior angles measure 90°, and the parallelogram is a *rectangle* , a four-sided plane polygon whose interior angles are all congruent (Fig. 4-24). The converse of this is also true: If a quadrilateral is a rectangle, then any given interior angle has a measure of 90°.

**Rectangle Diagonals**

Suppose that we have a parallelogram defined by four points *P, Q, R* , and *S* . Let *D* be the diagonal connecting *P* and *R* ; let *E* be the diagonal connecting *Q* and *S* . Let the length of *D* be denoted by *d* ; let the length of *E* be denoted by *e* (Fig. 4-25). If *d* = *e* , then the parallelogram is a rectangle. The converse is also true: If a parallelogram is a rectangle, then *d* = *e* . Thus a parallelogram is a rectangle if and only if its diagonals have equal lengths.

**Fig. 4-25** The diagonals of a rectangle have equal length.

**Rhombus Diagonals**

Suppose that we have a parallelogram defined by four points *P, Q, R* , and *S* . Let *D* be the diagonal connecting *P* and *R* ; let *E* be the diagonal connecting *Q* and *S* . If *D* is perpendicular to *E* , then the parallelogram is a *rhombus* , a four-sided plane polygon whose sides are all equally long (Fig. 4-26). The converse is also true: If a parallelogram is a rhombus, then *D* is perpendicular to *E* . Thus a parallelogram is a rhombus if and only if its diagonals are perpendicular.** **

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