Quadrilaterals • Topic 2 of 3

Special Parallelograms: Rectangle, Square, & Rhombus

What are rectangles, squares, and rhombuses? Rectangles, squares, and rhombuses are special members of the parallelogram family. Because they are parallelograms, they automatically inherit all regular parallelogram features (like opposite sides being equal and diagonals bisecting each other). However, each has unique "superpowers" that set them apart.

Rectangle: A rectangle is a parallelogram with four perfect right angles (\(90^\circ\) each). Think of a standard brick, a smartphone screen, or a doorway.

  • Special Superpower: Its diagonals are completely equal in length.

Rhombus: A rhombus is a parallelogram where all four sides are exactly equal in length. Think of a diamond pattern on a playing card or a kite shape with equal edges.

  • Special Superpower: Its diagonals cross each other at a perfect \(90^\circ\) right angle. We say they are perpendicular bisectors of each other.

Square: A square is the ultimate geometric shape because it is completely regular. It is a perfect hybrid combination of a rectangle and a rhombus! It has four equal sides AND four right angles. Think of a slice of sandwich bread or a tile on the floor.

  • Special Superpowers: It has all properties of a rectangle and rhombus combined: all sides equal, all angles \(90^\circ\), diagonals equal in length, and diagonals crossing at exactly \(90^\circ\).
ShapeParallel SidesAll Sides Equal?All Angles \(90^\circ\)?Diagonal Relationship
Parallelogram2 PairsNoNoBisect each other
Rectangle2 PairsNoYesBisect each other + Equal in length
Rhombus2 PairsYesNoBisect each other + Perpendicular (\(90^\circ\))
Square2 PairsYesYesBisect each other + Equal + Perpendicular
Special ParallelogramsRectangleAll angles=90°Equal diagonalsSquareAll sides equalAll angles=90°RhombusAll sides equalDiagonals bisect at 90°Diagonals equalDiagonals ⊥Hierarchy: Square ⊂ Rectangle ⊂ Parallelogram ⊂ QuadrilateralSquare is a special rectangle AND a special rhombus
Solved Examples
1
Worked Example
In a rectangle \(XYZW\), the diagonal \(XZ\) has a length of \(5x - 3\text{ cm}\), and the diagonal \(YW\) has a length of \(2x + 6\text{ cm}\). Find the value of \(x\) and the actual length of the diagonals.
Solution*Step 1: Recall the unique property of rectangles: the diagonals are completely equal in length. *Step 2: Set the two expressions equal to each other:* \(5x - 3 = 2x + 6\) *Step 3: Subtract \(2x\) from both sides of the equation:* \(3x - 3 = 6\) *Step 4: Add \(3\) to both sides:* \(3x = 9\) *Step 5: Divide by \(3\):* \(x = 3\) *Step 6: Substitute \(x = 3\) back into either diagonal expression to find the length:* \(\text{Length} = 2(3) + 6 = 6 + 6 = 12\text{ cm}\)

Key Points

  • A rectangle has four right angles (\(90^\circ\)) and diagonals that are equal in length.
  • A rhombus has four equal sides and diagonals that intersect at right angles (\(90^\circ\)).
  • A square satisfies every rule for both rectangles and rhombuses simultaneously.
  • Every square is a rectangle and a rhombus, but not every rectangle or rhombus is a square.
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Quick Check — 4 MCQs
Tap an option to check your answer0 / 4
Q1.Each angle of a rectangle is:
Explanation: $90^\circ$.
Q2.A rhombus has:
Explanation: All sides equal.
Q3.A square is both a rectangle and a:
Explanation: Rhombus.
Q4.The diagonals of a rhombus are:
Explanation: Perpendicular.
Practice more MCQs → 📝 Assignment · 35 marks