B2. Consider two circles with centers $S_{1}$ and $S_{2}$. The centers of these two circles are $21 \mathrm{~cm}$ apart, and their common chord is $24 \mathrm{~cm}$ long. The radius of the first circle is $13 \mathrm{~cm}$. Draw a sketch. Calculate the radius of the second circle and the perimeter of the shape that represents the union of the two circles.
B2. We connect the centers of both circles with both endpoints of the chord. The radius of the first circle, half of the chord $t$, and the segment $d_{1}$ from the center of the first circle $S_{1}$ to the midpoint of the chord form a right triangle. Using the Pythagorean theorem, we get $d_{1}=5 \mathrm{~cm}$. Since the distance between the centers of both circles is $21 \mathrm{~cm}$, the distance from the center of the second circle to the midpoint of the chord is $d_{2}=21-5=16 \mathrm{~cm}$. We use the Pythagorean theorem again and get the radius of the second circle $r_{2}=20 \mathrm{~cm}$. To calculate both central angles, we use trigonometric functions, for example: $\sin \frac{\varphi_{1}}{2}=\frac{t / 2}{r_{1}}$. We get $\varphi_{1} \doteq 134.76^{\circ}$ and $\varphi_{2} \doteq 73.74^{\circ}$. The corresponding arc lengths are $l_{1}=\frac{\varphi_{1}}{180^{\circ}} \cdot \pi \cdot r_{1} \doteq 30.58 \mathrm{~cm}$ and $l_{2} \doteq 25.74 \mathrm{~cm}$. The perimeter of our shape is the sum of the circumferences of both circles minus the lengths of both arcs $o=o_{1}+o_{2}-l_{1}-l_{2} \doteq 151.03 \mathrm{~cm}$.

Sketch of both circles with the chord marked 1 point

Calculation of the distance from $S_{1}$ to the chord: $d_{1}=5 \mathrm{~cm}$ 1 point

Calculation of $r_{2}=20 \mathrm{~cm}$ 1 point

![](https://cdn.mathpix.com/cropped/2024_06_07_9fbd0ae098c18b73b6bfg-21.jpg?height=55&width=1725&top_left_y=2474&top_left_x=177)

![](https://cdn.mathpix.com/cropped/2024_06_07_9fbd0ae098c18b73b6bfg-21.jpg?height=48&width=1725&top_left_y=2529&top_left_x=177)

![](https://cdn.mathpix.com/cropped/2024_06_07_9fbd0ae098c18b73b6bfg-21.jpg?height=48&width=1725&top_left_y=2580&top_left_x=177)

![](https://cdn.mathpix.com/cropped/2024_06_07_9fbd0ae098c18b73b6bfg-21.jpg?height=51&width=1725&top_left_y=2627&top_left_x=177)

![](https://cdn.mathpix.com/cropped/2024_06_07_9fbd0ae098c18b73b6bfg-21.jpg?height=54&width=1716&top_left_y=2674&top_left_x=176)