Question 4

We will determine the properties of the diagonals of the new shape to identify the quadrilateral.

Step 1 — Understand the original figure

The diagram shows two line segments. These segments act as the diagonals of a quadrilateral. Let us call the vertical segment $D_V$ and the horizontal segment $D_H$. We can count the units between the dots. The length of $D_V$ is 5 units. The length of $D_H$ is 5 units. So, the diagonals are equal in length ($D_V = D_H$). They intersect at the exact center. This means they bisect each other. They are also perpendicular to each other. A quadrilateral whose diagonals are equal, bisect each other, and are perpendicular is a square.

Step 2 — Modify one diagonal

We extend one of the diagonals on both sides by 2 cm. Let us choose the horizontal diagonal, $D_H$. Its original length was 5 units. The new length of the horizontal diagonal, $D_H'$, will be: $D_H' = 5 \text{ units} + 2 \text{ cm} + 2 \text{ cm}$ $D_H' = 5 \text{ units} + 4 \text{ cm}$ The vertical diagonal, $D_V$, remains unchanged. Its length is still 5 units. The diagonals still intersect at the same central point. This means they still bisect each other. They are also still perpendicular to each other. Now, the length of $D_V$ is 5 units and the length of $D_H'$ is $5 \text{ units} + 4 \text{ cm}$. Since $5 \text{ units} \neq 5 \text{ units} + 4 \text{ cm}$, the diagonals are no longer equal in length.

Step 3 — Identify the new quadrilateral

Let us list the properties of the diagonals of the new quadrilateral. The diagonals bisect each other. The diagonals are perpendicular to each other. The diagonals are of unequal length. A quadrilateral whose diagonals bisect each other is a parallelogram. A parallelogram whose diagonals are perpendicular is a rhombus. Since the diagonals are not equal, it is not a square or a rectangle. Therefore, the new quadrilateral formed is a rhombus.

Answer

The quadrilateral you will get now is a rhombus.

Justification: The diagonals of the new quadrilateral still bisect each other and are perpendicular. However, they are no longer equal in length. A quadrilateral with diagonals that bisect each other and are perpendicular is a rhombus.