The Rectangle and the Square

The Rectangle and the Square — Moosa Academy

A rectangle is not a separate kind of shape from a parallelogram — it is a parallelogram with one extra condition. Add another condition and you get a square. Seeing them as a nested family, rather than three unrelated shapes, makes every property easy to recall.

Concept A family, not three shapes

Parallelogram — opposite sides parallel and equal; diagonals bisect each other.

Rectangle — a parallelogram whose angles are all 90°.

Square — a rectangle whose sides are all equal.

\text{square} \subset \text{rectangle} \subset \text{parallelogram}

Reading that chain from left to right: every square is a rectangle, and every rectangle is a parallelogram. Reading it backwards is not allowed — most parallelograms are not rectangles.

Concept The three shapes side by side

All three have diagonals that bisect each other, because all three are parallelograms. What changes is the extra behaviour of those diagonals.

Theorem A rectangle has equal diagonals

In rectangle $ABCD$ , compare triangles $ABC$ and $BAD$ :

AB

is shared by both

BC = AD

— opposite sides of a parallelogram

\angle ABC = \angle BAD = 90^\circ

— it is a rectangle

The two triangles are congruent by SAS, so their third sides match:

AC = BD

This is the property that distinguishes a rectangle: right angles force the two diagonals to have the same length.

Example Using the diagonal of a rectangle

A rectangle measures 6 cm by 8 cm, and its diagonals meet at $M$ . Find the length of $AM$ .

The diagonal is the hypotenuse of a right triangle with legs 6 and 8

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

M

bisects it, so

AM = 10 \div 2 = 5 \text{ cm}

Since the diagonals are equal and bisect each other, all four segments from $M$ to the corners measure 5 cm — so $M$ is the centre of a circle through all four vertices.

Note The three compared

Property	Parallelogram	Rectangle	Square
Angles	Opposite equal	All 90°	All 90°
Sides	Opposite equal	Opposite equal	All equal
Diagonals	Bisect each other	Also equal	Also perpendicular

Each column keeps everything to its left and adds one more line. A square is simply a shape that satisfies every row at once.

Summary

A rectangle is a parallelogram with all angles 90°.
A square is a rectangle with all sides equal.
Every square is a rectangle, and every rectangle is a parallelogram.
All three have diagonals that bisect each other.
A rectangle's diagonals are also equal in length.
A square's diagonals are equal, bisecting, and perpendicular.