Properties of a Rhombus

Properties of a Rhombus

How to solve the rhombus problems by using its properties: definition, properties of its diagonals and angles, examples, and their solutions.

Definition

A rhombus is a parallelogram whose sides are all congruent.

A rhombus is a parallelogram
whose sidea are all congruent.

So a rhombus
has all the properties of a parallelogram.

Properties of a parallelogram

Property: Diagonals

For a rhombus, the diagonals perpendicularly bisect each other.

For a rhombus,
the diagonals perpendicularly bisect each other.

Example 1

If ABCD is a rhombus, find BC and BD. AD = 5, PC = 3.

The given figure is a rhombus.
So its sides are all congruent.

AD = 5

So BC = 5.

The diagonals perpendicularly bisect each other.

So ∠CPB is a right angle.

See △PBC.

It's a right triangle.
And starting from the shortest side,
the sides are (3, PB, 5).

So △PBC is a (3, 4, 5) right triangle.

Pythagorean triples

So PB = 4.

DP = PB = 4

So BD = 4 + 4
= 8.

Property: Angles

For a rhombus, the opposite interior angles are congruent.

A rhombus is also a parallelogram.

So, for a rhombus,
its opposite interior angles are congruent.

Properties of a parallelogram - Angles

For a rhombus, the diagonals bisect the interior angles.

For a rhombus,
the diagonals bisect the interior angles.

Example 2

Find the measure of angle ABC and the measure of angle PAB. [measure of angle ADP] = 50

The given figure is a rhombus.

ADC and ∠ABC are the opposite angles.
So these two angles are congruent.

Next, its diagonal DB bisects ∠ADC and ∠ABC.

So ∠[blue dot] are all congruent.

m∠[blue dot] = 50.

So m∠ABC = 2⋅m∠[blue dot]
= 2⋅[50]
= 100.

The other diagonal AC bisects ∠DAB.

For a rhombus,
the diagonals perpendicularly bisect each other.

So AC and DB perpendicularly bisect each other.

So ∠DPA is a right angle.

DPA is a triangle.

m∠[blue dot] = 50
m∠DPA = 90

So m∠[green dot] + [50] + [90] = 180.

Interior angles of a triangle

50 + 90 = 140

Move +140 to the right side.

Then m∠[green dot] = 40.

m△PAB = m∠[green dot] = 40

So m△PAB = 40.