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25^2 = 7^2 + 24^2 \quad \Rightarrow \quad 625 = 49 + 576 \quad \Rightarrow \quad 625 = 625
The triangle is confirmed as a right triangle. The area \(A\) of the triangle is:
A = \frac{1}{2} \times 7 \times 24 = 84 \text{ cm}^2
The semi-perimeter \(s\) is:
s = \frac{7 + 24 + 25}{2} = 28 \text{ cm}
The radius \(r\) of the inscribed circle is:
r = \frac{A}{s} = \frac{84}{28} = 3 \text{ cm}
\text{Area} = \pi r^2 = \pi \times 3^2 = 9\pi \text{ cm}^2
Thus, the area of the inscribed circle is \(\boxed{9\pi}\).
An equilateral triangle has a side length of 10 cm. If each side is decreased by 2 cm, by how many square centimeters does the area decrease?