Rectangle, rhombus, and square

Once you know everything about parallelograms, the rectangle, rhombus, and square come almost for free , each just adds one more requirement. Understanding which one adds which is the secret to never confusing them again.

Concept

Rectangle = parallelogram with a right angle.

If one angle is $90^\circ$, then because adjacent angles are supplementary all four must be $90^\circ$. So a rectangle has four right angles.

Extra property: The diagonals of a rectangle are equal in length (in addition to bisecting each other).

Why? The two triangles formed by a diagonal are congruent right triangles with the same hypotenuse , so by Pythagoras the two diagonals must have the same length.

Rhombus = parallelogram with all sides equal.

So all four sides of a rhombus have the same length. (Some people call it a "diamond".)

Extra property: The diagonals of a rhombus are perpendicular to each other (in addition to bisecting each other). They also bisect the angles of the rhombus.

Why? Triangle $\triangle AOB$ in a rhombus has $OA = OC$ and $OB = OD$ (from parallelogram) plus $AB = AD$ (rhombus sides). This forces $\triangle AOB \cong \triangle AOD$, and the equal angles at $O$ that share a line must each be $90^\circ$.

Square = rectangle + rhombus.

A square has all the properties of both shapes:

• Four equal sides (from rhombus).
• Four right angles (from rectangle).
• Diagonals equal and perpendicular and bisecting each other.

A square is the most symmetric quadrilateral. It has $4$ axes of symmetry and rotational symmetry of order $4$.

Family tree summary.

                Quadrilateral
                     |
                Parallelogram
                /        \
        Rectangle      Rhombus
                \        /
                 Square

To identify a quadrilateral from a description, ask:

1. Are opposite sides parallel? If yes, parallelogram.
2. Is there a right angle? If yes, also rectangle.
3. Are all sides equal? If yes, also rhombus.
4. Both? It is a square.

Equivalently, look at the diagonals:

Worked examples

Example 1. In rectangle $ABCD$, $AB = 8$ cm and $BC = 6$ cm. Find diagonal $AC$.

• By Pythagoras: $AC = \sqrt{8^2 + 6^2} = \sqrt{100} = 10$ cm.

Example 2. In rhombus $PQRS$, one diagonal is $24$ cm and the other is $10$ cm. Find each side.

• Diagonals bisect each other at right angles, forming four right triangles.
• Each side is the hypotenuse of a right triangle with legs $12$ and $5$.
• Side $= \sqrt{12^2 + 5^2} = \sqrt{169} = 13$ cm.

Example 3. A square has diagonal $14\sqrt{2}$ cm. Find its side and area.

• If side $= s$, diagonal $= s\sqrt{2} = 14\sqrt{2}$, so $s = 14$ cm.
• Area $= 14^2 = 196$ cm$^2$.

Example 4. The diagonals of a quadrilateral are equal and bisect each other but are not perpendicular. Identify the shape.

• "Bisect each other" $\Rightarrow$ parallelogram.
• "Equal" $\Rightarrow$ rectangle.
• "Not perpendicular" $\Rightarrow$ not a rhombus, hence not a square.
• The shape is a (non-square) rectangle.

Try it yourself

1. The diagonals of a rectangle are $10$ cm long. If one side is $6$ cm, find the other side.
2. The diagonals of a rhombus are $16$ cm and $30$ cm. Find each side.
3. The area of a square is $144$ cm$^2$. Find its diagonal.
4. A quadrilateral has all four sides $5$ cm and all four angles $90^\circ$. What is it?
5. Show that the diagonal of a square is $s\sqrt{2}$ where $s$ is the side.
6. Decide whether a parallelogram with one diagonal perpendicular to a side is necessarily a rectangle.
7. List one property a rectangle has that a parallelogram (in general) does not.
8. List one property a rhombus has that a parallelogram (in general) does not.

Activity

Diagonal detective. Draw a parallelogram, a rectangle, a rhombus, and a square , each on a separate sheet. Measure both diagonals of each and record. Then measure the angle between them. Make a $4 \times 3$ table: (shape, are diagonals equal?, are they perpendicular?). The table is the "diagonal fingerprint" of each shape and matches the summary above.