Question 2

We can make a square with half the area of the first square by folding its corners inwards.

Step 1 — Prepare the original square

Let the side length of the original square be $s$. Its area is $A_1$. $A_1 = s \times s$

$\boxed{A_1 = s^2 \text{ square units}}$

Take a square piece of paper. To prepare for folding, we need to find its center. Fold the square in half horizontally, then unfold it. Fold the square in half vertically, then unfold it. The creases will cross at the exact center of the square. These creases also mark the midpoints of each side of the square.

Step 2 — Fold the corners to form the new square

Now, fold each of the four corners of the original square inwards. Each corner should be folded so that its vertex meets the center of the square. The fold lines will connect the midpoints of adjacent sides. For example, the bottom-left corner is folded along a line connecting the midpoint of the bottom side to the midpoint of the left side. After folding all four corners, the resulting shape is a new square. This new square has vertices at the midpoints of the original square's sides. Let's call the area of this new square $A_2$.

Step 3 — Calculate the area of the new square

When we fold the corners, we are removing four identical right-angled triangles from the original square. Each of these triangles has two equal sides (legs). The length of each leg is half the side length of the original square. So, each leg has a length of $s/2$. The area of one such right-angled triangle is calculated as: $\text{Area of one triangle} = \frac{1}{2} \times \text{base} \times \text{height}$ $= \frac{1}{2} \times \frac{s}{2} \times \frac{s}{2}$ $= \frac{s^2}{8}$ There are four such triangles that are folded away. The total area removed from the original square is: $\text{Total area removed} = 4 \times \frac{s^2}{8}$ $= \frac{4s^2}{8}$ $= \frac{s^2}{2}$ The area of the new square ($A_2$) is the area of the original square ($A_1$) minus the total area removed. $A_2 = A_1 - \text{Total area removed}$ $= s^2 - \frac{s^2}{2}$

$\boxed{A_2 = \frac{s^2}{2} \text{ square units}}$ The area of the new square is half the area of the first square.

Answer

(i) Cut out a square from a piece of paper. (ii) Fold the square in half horizontally and vertically to find its center and the midpoints of its sides. (iii) Fold each of the four corners of the square inwards so that each corner vertex meets the center of the square. The resulting shape is a new square whose area is half the area of the original square.