Lesson 9.2.4

Properties of Rhombi

A rhombus is a parallelogram with four congruent sides. Its diagonals are perpendicular and each bisects a pair of opposite angles.

Introduction

Rhombus: all sides equal, diagonals perpendicular

Past Knowledge

Parallelogram properties (9.2.1). Rectangle (9.2.3). Perpendicular lines.

Today's Goal

Prove and apply rhombus properties: perpendicular diagonals and angle bisection.

Future Success

Squares (9.2.5), area via diagonals, kites (9.3.3).

Key Concepts

Rhombus = Parallelogram + Four Equal Sides

A rhombus has ALL parallelogram properties, plus:

All four sides are congruent
Diagonals are perpendicular
Each diagonal bisects a pair of opposite angles

Area of a Rhombus

where and are the diagonals

Theorem & Proof

Two-Column Proof: Diagonals of a Rhombus Are Perpendicular

Given: Rhombus with diagonals intersecting at

Prove:

#	Statement	Reason
1		Definition of rhombus (all sides congruent)
2		Diagonals of a parallelogram bisect each other
3		Reflexive Property
4		SSS Congruence (steps 1, 2, 3)
5		CPCTC
6	→	Linear pair of congruent angles → each is 90°

∎ Equal sides + bisected diagonals → SSS → congruent angles at center → must be 90°.

Worked Examples

Basic

Finding Side from Diagonals

Rhombus with diagonals 16 and 12. Find the side length.

Half-diagonals: 8 and 6. Diagonals are ⊥ → right triangle.

Side = 10

Intermediate

Area from Diagonals

Rhombus with diagonals 24 and 10. Find the area.

Area = 120 square units

Advanced

Angle Bisection

Rhombus ABCD has ∠ABC = 120°. Find ∠ABD.

Diagonal BD bisects ∠B:

∠ABD = 60°

Common Pitfalls

Thinking Diagonals Are Congruent

In a rhombus, diagonals are perpendicular but NOT congruent (unless it's a square). Congruent diagonals = rectangle property.

Confusing Rhombus and Diamond

A “diamond” shape on a playing card is close to a rhombus but may not be one. A rhombus must be a parallelogram (opposite sides parallel).

Real-Life Applications

Baseball Diamonds

A baseball “diamond” is actually a square (a special rhombus) with 90-foot sides. The perpendicular diagonals explain why the distance from home plate to second base and from first to third are different.

Quilting

Many quilt patterns use rhombus-shaped fabric pieces that tessellate beautifully. The angle-bisection property helps quilters calculate precise cutting angles.