[Pythagorean Theorem (direct and inverse).]  [Tangent properties and criteria]

Circles with radii 8 and 3 touch each other internally. A tangent is drawn from the center of the larger circle to the smaller circle. Find the length of this tangent.
The line connecting the centers of two tangent circles passes through their point of tangency.

## Solution

Let $O$ and $O_{1}$ be the centers of circles with radii 8 and 3, respectively, and $A$ be the point of tangency, $O M-$ be the desired tangent. Then

$$
O O_{1}=O A-O O_{1}=8-3=5
$$

Therefore,

$$
O M^{2}=O O^{2}_{1}-O_{1} M^{2}=25-9=16, O M=4
$$

![](https://cdn.mathpix.com/cropped/2024_05_06_9e7ee8b57660b9721937g-12.jpg?height=449&width=506&top_left_y=2382&top_left_x=776)

## Answer

![](https://cdn.mathpix.com/cropped/2024_05_06_9e7ee8b57660b9721937g-13.jpg?height=177&width=1326&top_left_y=166&top_left_x=0)

How many parts do $n$ planes divide space when they all pass through one point, and no three have a common line?