To solve for the perimeter of the smaller square, let’s analyze the geometry:

1. The side length of the larger square is $x = 25$ cm.
2. The smaller square has its vertices touching the midpoints of the sides of the larger square.

Step-by-Step Solution

1. Determine the Side Length of the Smaller Square: Since the smaller square is rotated within the larger square, its diagonal is equal to the side length of the larger square (25 cm).

   Let $s$ represent the side length of the smaller square. The relationship between the side length $s$ and the diagonal $d$ of a square is given by: $d = s \sqrt{2}$

   Setting $d = 25$ cm, we get: $s \sqrt{2} = 25$

   Solving for $s$: $s = \frac{25}{\sqrt{2}} = \frac{25 \sqrt{2}}{2} \approx 17.68 \text{ cm}$

2. Calculate the Perimeter of the Smaller Square: The perimeter $P$ of a square is given by: $P = 4s$ Substituting $s \approx 17.68$ cm: $P \approx 4 \times 17.68 = 70.7 \text{ cm}$

Answer

The perimeter of the smaller square is approximately 70.7 cm.

Would you like further details or have any questions?

Related Questions

1. How would the perimeter change if the side length of the larger square increased?
2. Can we use trigonometry to verify the side length of the smaller square?
3. How do you calculate the area of the smaller square?
4. What is the relationship between the areas of the two squares?
5. How does the diagonal of a square relate to its side length in general?

Tip

For squares inscribed at a 45-degree angle within another square, the diagonal of the smaller square equals the side of the larger square.