Quadrilaterals | A

Question 1

Are squares the only quadrilaterals that have equal sidelengths? Let us explore this question through construction.

Draw two equal sides $AD$ and $AB$ , that are not perpendicular to each other.

Solution

Understand the Question

A quadrilateral with four sides of equal length is a rhombus.
A square is a specific type of rhombus where all four interior angles are right angles ( $90^\circ$ ).
By constructing a quadrilateral with equal sides where the angle between adjacent sides is not $90^\circ$ (e.g., $50^\circ$ ), we create a rhombus that is not a square.

Step 1 · Draw Two Equal Adjacent Sides

Draw two equal segments $AD$ and $AB$ meeting at vertex $A$ at an angle of $50^\circ$ (which is not $90^\circ$ ). Diagram 1

Step 2 · Draw an Arc from Vertex D

With $D$ as the centre and radius equal to $AD$ , draw an arc to mark the possible positions for vertex $C$ . Diagram 2

Step 3 · Locate Vertex C

With $B$ as the centre and radius equal to $AB$ , draw another arc intersecting the first arc at point $C$ . Diagram 3

This ensures $BC = AB$ and $DC = AD$ .

Step 4 · Complete the Quadrilateral

Join point $C$ to point $D$ and point $C$ to point $B$ to complete the quadrilateral $ABCD$ . Diagram 4

Step 5 · Analyze the Quadrilateral

From the construction: $AB = BC = CD = DA$ $\angle A = 50^\circ \neq 90^\circ$

A quadrilateral with all four sides equal is a rhombus. Since its interior angles are not $90^\circ$ , it is a rhombus and not a square.

Answer

No, squares are not the only quadrilaterals with equal side lengths; a rhombus also has four equal sides.

Common Mistakes

Equating Equal Sides Exclusively with Squares: Assuming every four-sided figure with equal sides must be a square. A square requires both equal sides and four right angles ( $90^\circ$ ).
Overlooking Angle Variation: Forgetting that changing the angle between adjacent sides produces a general rhombus while keeping all side lengths equal.

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