Extend one of the diagonals on both sides by $2\text{ cm}$ .

What quadrilateral will you get now? Justify your answer.

Question

Extend one of the diagonals on both sides by . What quadrilateral will you get now? Justify your answer.

Homi · Accepted Answer

Understand the Question

The type of a quadrilateral can be determined by the properties of its diagonals (length, intersection angle, and whether they bisect each other).
In the original figure, the two diagonals are equal in length, perpendicular, and bisect each other, forming a square.
Extending one diagonal equally on both sides preserves perpendicularity and mutual bisection, but makes the diagonal lengths unequal.

Step 1 · Analyze the Original Diagonals

From the given dot grid, let the vertical diagonal be $D_V$ and the horizontal diagonal be $D_H$ . Diagram 1

The diagonals are equal in length ( $D_V = D_H$ ), bisect each other, and are perpendicular to each other.

Step 2 · Modify One Diagonal

Extend the horizontal diagonal $D_H$ by $2\text{ cm}$ on both sides. Diagram 2

\begin{aligned} D_H' &= 5\text{ units} + 2\text{ cm} + 2\text{ cm} \\[0.6em] &= 5\text{ units} + 4\text{ cm} \end{aligned}

The vertical diagonal remains $D_V = 5\text{ units}$ .

Since $5\text{ units} \neq 5\text{ units} + 4\text{ cm}$ , the diagonals are now unequal ( $D_V \neq D_H'$ ).

Step 3 · Identify the New Quadrilateral

The diagonals of the modified quadrilateral satisfy:

They bisect each other (extended equally on both sides).
They are perpendicular to each other (angle between them is unchanged at $90^\circ$ ).
They are unequal in length ( $D_H' > D_V$ ).

A quadrilateral whose diagonals bisect each other at right angles with unequal lengths is a rhombus.

Answer

The new quadrilateral is a rhombus.

Justification: The diagonals still bisect each other perpendicularly at right angles, but are no longer equal in length.

Common Mistakes

Extending Only One Side: Extending a diagonal on only one side shifts the intersection point, meaning the diagonals would no longer bisect each other.
Confusing Square and Rhombus: A square requires equal perpendicular diagonals. When the diagonals become unequal while remaining perpendicular bisectors, the figure becomes a rhombus (not a square or rectangle).

Question 4