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Quadrilaterals for Physics Help

By — McGraw-Hill Professional
Updated on Sep 17, 2011

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

Basics of Geometry Quadrilaterals Parallelogram Diagonals

Fig. 4-22 Triangles defined by the minor diagonal ( a ) or the major diagonal ( b ) of a parallelogram are congruent.

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:

Basics of Geometry Quadrilaterals Bisection Of Parallelogram Diagonals

Fig. 4-23 The diagonals of a parallelogram bisect each other.

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°.

Basics of Geometry Quadrilaterals Rectangle

Fig. 4-24 If a parallelogram has one right interior angle, then the parallelogram is a rectangle.

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.

Basics of Geometry Quadrilaterals Rhombus Diagonals

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.

Basics of Geometry Quadrilaterals Rhombus Diagonals

Fig. 4-26 The diagonals of a rhombus are perpendicular.

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