## Task B-2.4.

The centers of two circles are $44 \mathrm{~cm}$ apart. If the radii of these circles are $15 \mathrm{~cm}$ and $37 \mathrm{~cm}$, what is the length of their common chord?
## First Solution.

Let's introduce the notation as shown in the figure.

![](https://cdn.mathpix.com/cropped/2024_05_30_ab4e7c3a83d03c059773g-09.jpg?height=663&width=854&top_left_y=565&top_left_x=521)

Let's calculate the area of triangle \( A S_{1} S_{2} \) using Heron's formula. For \( a = |S S_{1}| = 44 \, \text{cm} \), \( b = |S_{2} A| = 37 \, \text{cm} \), and \( c = |S_{1} A| = 15 \, \text{cm} \), we get \( s = \frac{1}{2}(15 + 37 + 44) = 48 \, \text{cm} \),

\[
P = \sqrt{s(s-a)(s-b)(s-c)} = \sqrt{48 \cdot 4 \cdot 11 \cdot 33} = 4 \cdot 3 \cdot 2 \cdot 11 = 264 \, \text{cm}^2
\]

On the other hand, the area of triangle \( A S_{1} S_{2} \) is \( P = \frac{1}{2} \cdot |S_{1} S_{2}| \cdot |A E| \).

From \( 264 = \frac{1}{2} \cdot 44 \cdot |A E| \), it follows that \( |A E| = 12 \, \text{cm} \).

The length of the common chord of these two circles is \( |A B| = 2 |A E| = 24 \, \text{cm} \).