The image shows a geometric figure consisting of three squares arranged around a right triangle. The squares are labeled with their areas:

1. The largest square has an area of $1369 \, \text{cm}^2$.
2. The smallest square has an area of $144 \, \text{cm}^2$.
3. The shaded square (whose area we need to find) is adjacent to these two squares.

Since the squares are arranged around a right triangle, this setup follows the Pythagorean theorem. The area of the shaded square corresponds to the square on the remaining side of the right triangle.

Solution:

Let's denote:

• The side length of the largest square as $c$.
• The side length of the smallest square as $a$.
• The side length of the shaded square as $b$.

According to the Pythagorean theorem: $a^2 + b^2 = c^2$

Given:

• $c^2 = 1369 \, \text{cm}^2 \Rightarrow c = \sqrt{1369} = 37 \, \text{cm}$
• $a^2 = 144 \, \text{cm}^2 \Rightarrow a = \sqrt{144} = 12 \, \text{cm}$

Now we can find $b^2$: $b^2 = c^2 - a^2$ $b^2 = 1369 - 144 = 1225$

Therefore: $b = \sqrt{1225} = 35 \, \text{cm}$

The area of the shaded square is $b^2$, which is: $1225 \, \text{cm}^2$

Answer:

The area of the shaded square is $\mathbf{1225 \, cm^2}$.

Would you like more details or have any questions?

Here are some related questions for further exploration:

1. How does the Pythagorean theorem apply to other shapes beyond squares?
2. What are some real-world applications of the Pythagorean theorem?
3. How can you use the side lengths to find the perimeter of the triangle?
4. Can the Pythagorean theorem be applied to non-right triangles?
5. How do you calculate the area of a square given its diagonal length?

Tip: Remember that the Pythagorean theorem only applies to right-angled triangles, so the relationship $a^2 + b^2 = c^2$ only holds in these cases.