Two circles are drawn, with the same center. A chord of the large circle is drawn, so that it is tangent to the small circle. If the chord has a length of $10,$ then find the area of the ring-shaped region that is inside the large circle but outside the small circle.

[asy]
unitsize(0.5 cm);

fill(Circle((0,0),3)^^Circle((0,0),2),evenodd+gray(0.75)+opacity(0.75));

draw(Circle((0,0),2));
draw(Circle((0,0),3));
draw((-sqrt(5),2)--(sqrt(5),2));

label("$10$", (0,2), N, red);
[/asy]


We are given two concentric circles (circles with the same center) and a chord of the larger circle, which is tangent to the smaller circle. The length of the chord is given as 10. Our task is to find the area of the ring-shaped region that is inside the larger circle but outside the smaller circle.

### Step 1: Define the radii of the circles
Let the radius of the larger circle be $ R $ and the radius of the smaller circle be $ r $.

### Step 2: Use the geometry of the problem
The chord of the larger circle is tangent to the smaller circle, and it has a length of 10. Since the two circles share the same center, the distance from the center to the midpoint of the chord will be the radius of the smaller circle, $ r $.

Additionally, the perpendicular from the center to the chord bisects the chord into two equal segments of 5 each (since the chord has a total length of 10).

### Step 3: Apply the Pythagorean theorem
We now have a right triangle where:
- One leg is the radius of the smaller circle, $ r $,
- The other leg is half of the chord's length, 5,
- The hypotenuse is the radius of the larger circle, $ R $.

By the Pythagorean theorem:
\[
R^2 = r^2 + 5^2
\]
\[
R^2 = r^2 + 25
\]

### Step 4: Calculate the area of the ring
The area of the ring-shaped region is the difference between the area of the larger circle and the area of the smaller circle. The area of a circle is given by $ \pi r^2 $, so:

\[
\text{Area of the ring} = \pi R^2 - \pi r^2 = \pi (R^2 - r^2)
\]

From the equation $ R^2 = r^2 + 25 $, we substitute this into the area formula:

\[
\text{Area of the ring} = \pi (r^2 + 25 - r^2) = \pi \times 25
\]

Thus, the area of the ring-shaped region is:

\[
\boxed{25\pi}
\]

This is the area of the ring-shaped region that is inside the large circle but outside the small circle.

### Would you like more details or have any questions?

Here are some related questions for further exploration:
1. What if the length of the chord was different? How would the area change?
2. How would the situation change if the circles were not concentric?
3. What if the chord was not tangent to the smaller circle, but intersected it? 
4. How does the position of the chord affect the area of the ring-shaped region?
5. Can you generalize the formula for the area of the ring if the chord length is given as $ L $?

Tip: Understanding how chords relate to the geometry of circles can greatly simplify solving circle-based problems.

Two circles are drawn, with the same center. A chord of the large circle is drawn, so that it is tangent to the small circle. If the chord has a length of 10, then find the area of the ring-shaped region that is inside the large circle but outside the small circle.

Solve the geometry problem involving two concentric circles, where a chord of length 10 in the larger circle is tangent to the smaller circle. Learn how to apply the Pythagorean theorem and circle area formulas to find the area of the ring-shaped region.

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