To solve the problem, we need to find the areas of the two concentric circles and then compute the area of the ring-shaped region between them.
Let \( r \) be the radius of the small circle and \( R \) be the radius of the large circle. The given chord of the large circle is tangent to the small circle and has a length of 10.
Step 1: Relationship between the chord and the circles
Since the chord is tangent to the small circle, it will be at a distance equal to the radius of the small circle \( r \) from the center of the circles. Let the perpendicular distance from the center of the circle to the chord be \( r \).
Let \( d \) be the distance from the center to the chord along the radius of the large circle. By the properties of chords in a circle, we know that:
\[ d = \sqrt{R^2 - \left( \frac{L}{2} \right)^2} \]
where \( L \) is the length of the chord. We know the length \( L = 10 \), so:
\[ \frac{L}{2} = 5 \]
Substituting into the equation gives:
\[ d = \sqrt{R^2 - 5^2} = \sqrt{R^2 - 25} \]
Since this distance \( d \) must also equal the radius \( r \) of the small circle, we can set up the equation:
\[ r = \sqrt{R^2 - 25} \]
Step 2: Area calculations
The area of the large circle \( A_R \) is given by:
\[ A_R = \pi R^2 \]
The area of the small circle \( A_r \) is given by:
\[ A_r = \pi r^2 \]
To find the area of the ring-shaped region, we compute:
\[ \text{Area of the ring} = A_R - A_r = \pi R^2 - \pi r^2 = \pi (R^2 - r^2) \]
Step 3: Determining \( R^2 - r^2 \)
We know that:
\[ r^2 = R^2 - 25 \]
Thus,
\[ R^2 - r^2 = R^2 - (R^2 - 25) = 25 \]
Step 4: Computing the area of the ring
Substituting \( R^2 - r^2 = 25 \) into the area of the ring, we have:
\[ \text{Area of the ring} = \pi (R^2 - r^2) = \pi (25) = 25\pi \]
Final answer:
Hence, the area of the ring-shaped region that is inside the large circle but outside the small circle is
\[ \boxed{25\pi} \]