10.216. Two circles, with radii of 4 and 8, intersect at a right angle. Determine the length of their common tangent.
Solution.

Let $A$ be the point of intersection of the circles with centers $O$ and $O_{1}$ (Fig. 10.26), $O A=8 \text{ cm}, O_{1} A=4 \text{ cm}$, $M N$ - their common tangent. According to the problem, the circles intersect at a right angle, so the tangents to them at point $A$ are perpendicular to each other. Therefore, the radii drawn to the point of tangency are also perpendicular. From $\triangle O A O_{1}: O O_{1}^{2}=O A^{2}+O_{1} A^{2}=80$. Draw $O_{1} D \| M N$. Then $O_{1} D \perp O M, M N=O_{1} D, M D=N O_{1}=4, O D=$ $=O M-M D=4$. From $\triangle O D O_{1}: O_{1} D=\sqrt{O O_{1}^{2}-O D^{2}}=8$.

Answer: 8.

![](https://cdn.mathpix.com/cropped/2024_05_21_3edb7be591bff54ce350g-0727.jpg?height=470&width=475&top_left_y=80&top_left_x=142)

Fig. 10.27