A square piece of paper is folded three times along its diagonal to get an isosceles triangle whose equal sides are 15 cm. What is the area of the unfolded original piece of paper?

Options

A. 8,100 sq cm
B. 3,600 sq cm
C. 900 sq cm
D. 2,500 sq cm

Answer

To find the area of the original square piece of paper, we can follow these steps:

### Step 1: Understand the relationship between the triangle and the square
When the square is folded along its diagonal, it forms an isosceles triangle. The equal sides of this triangle are given as 15 cm.

### Step 2: Determine the side length of the square
Since the triangle is formed by folding the square along its diagonal, the length of each side of the square can be determined by the relationship between the side of the square (s) and the diagonal (d) of the square. The diagonal of a square is given by the formula:
\[ d = s\sqrt{2} \]
When the square is folded, the diagonal becomes the hypotenuse of the isosceles triangle formed.

In this case, the equal sides of the triangle are 15 cm, which means:
\[ d = 15 \text{ cm} \]

### Step 3: Calculate the side length of the square
We can rearrange the formula for the diagonal to find the side length of the square:
\[ s = \frac{d}{\sqrt{2}} \]
Substituting the value of d:
\[ s = \frac{15}{\sqrt{2}} \]
To simplify this, we can multiply the numerator and denominator by \(\sqrt{2}\):
\[ s = \frac{15\sqrt{2}}{2} \]

### Step 4: Calculate the area of the square
The area (A) of a square is given by:
\[ A = s^2 \]
Substituting the value of s:
\[ A = \left(\frac{15\sqrt{2}}{2}\right)^2 \]
\[ A = \frac{225 \cdot 2}{4} \]
\[ A = \frac{450}{4} \]
\[ A = 112.5 \text{ cm}^2 \]

### Step 5: Find the area of the original square
However, we need to remember that the triangle formed is half of the square when folded. Since the triangle is formed from half of the square, we need to multiply the area of the triangle by 2 to find the area of the original square.

Thus, the area of the original square is:
\[ A = 2 \times 112.5 = 225 \text{ cm}^2 \]

### Final Calculation
However, since we are looking for the area of the unfolded original piece of paper, we realize that we need to consider the equal sides of the triangle directly. The equal sides of the triangle are 15 cm, and since the triangle is formed by folding the square, the side length of the square is actually:
\[ s = 15 \times 2 = 30 \text{ cm} \]

Now, the area of the original square is:
\[ A = s^2 = 30^2 = 900 \text{ cm}^2 \]

### Conclusion
The area of the unfolded original piece of paper is **900 cm²**.