The Rhombus | Moosa Academy

The Rhombus — Moosa Academy

A rhombus is a parallelogram with all four sides equal. That one change forces its diagonals to behave quite differently: they meet at right angles and cut the corner angles exactly in half. Make the angles 90° as well and the rhombus becomes a square.

Concept What a rhombus inherits and what it adds

Every rhombus is a parallelogram, so opposite sides are parallel, opposite angles are equal, and the diagonals bisect each other at

M

All four sides equal:

AB = BC = CD = DA

Diagonals perpendicular:

AC \perp BD

Diagonals bisect the angles they pass through

Theorem Why the diagonals are perpendicular

Look at the diagonal $BD$ and the two ends of the other diagonal. Because all sides are equal:

AB = AD

, so

A

is the same distance from

B

and from

D

CB = CD

, so

C

is also the same distance from

B

and from

D

Two points that are each equidistant from $B$ and $D$ must both lie on the perpendicular bisector of $BD$ . The line through them is $AC$ , so:

AC \perp BD

The same argument, run the other way round, shows $BD$ is the perpendicular bisector of $AC$ . This is why the four small triangles inside a rhombus are all right-angled and congruent.

Example Finding a side from the diagonals

A rhombus has diagonals of 16 cm and 12 cm. Find the length of one side.

The diagonals bisect each other, so the half-diagonals are 8 cm and 6 cm

They are perpendicular, so those halves are the legs of a right triangle

\text{side} = \sqrt{8^2 + 6^2} = \sqrt{100} = 10 \text{ cm}

Both facts were needed: bisecting gave the halves, and perpendicularity allowed Pythagoras. In a general parallelogram this shortcut does not exist.

Concept When a rhombus becomes a square

A rhombus already has all sides equal. Add right angles and every remaining property of a square follows:

\text{rhombus} + 90^\circ \text{ angles} = \text{square}

The diagonals stay perpendicular, and now they are equal as well

A rhombus has two lines of symmetry, along its diagonals. A square has four, because the two lines through the midpoints of opposite sides become symmetry lines too.

Note Rhombus against square

Property	Rhombus	Square
Sides	All equal	All equal
Angles	Opposite equal	All 90°
Diagonals	Perpendicular	Perpendicular and equal
Lines of symmetry	2	4

The full chain of nesting reads $\text{square} \subset \text{rhombus} \subset \text{parallelogram}$ . A square is the only shape that is both a rhombus and a rectangle.

Summary

A rhombus is a parallelogram with all four sides equal.
It keeps every parallelogram property, including bisecting diagonals.
Its diagonals are perpendicular, because each lies on the perpendicular bisector of the other.
Its diagonals also bisect the angles they pass through.
Half-diagonals form a right triangle, so a side is √(half² + half²).
A rhombus with right angles is a square, which then also has equal diagonals.