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We are given a circle with a chord of length \( \sqrt{20} \, \text{cm} \), which is tangent to the smaller inner circle, and we need to find the area of the shaded region (the area between the outer and inner circles).

### Steps to solve the problem:

1. **Identify key parameters**:
   - The chord of the larger circle has a length of \( \sqrt{20} \, \text{cm} \).
   - This chord is tangent to the smaller circle, implying that the perpendicular distance from the center of the larger circle to the chord equals the radius of the inner circle.

2. **Using geometry**:
   The chord is tangent to the inner circle, and the perpendicular from the center of the larger circle to the chord bisects it. Let's denote:
   - \( R \) as the radius of the larger circle.
   - \( r \) as the radius of the smaller circle (inner circle).

   By the properties of a circle, the distance from the center of the larger circle to the chord is equal to the radius of the smaller circle \( r \). The midpoint of the chord will be at a distance \( r \) from the center, forming a right triangle with half of the chord as one leg and \( R - r \) as the other leg.

3. **Forming a right triangle**:
   From the center of the larger circle, draw a perpendicular to the chord. This divides the chord into two equal parts, each of length \( \frac{\sqrt{20}}{2} = \sqrt{5} \, \text{cm} \). The hypotenuse of this right triangle is \( R \) (the radius of the larger circle), and one leg is \( \sqrt{5} \), while the other leg is \( R - r \).

   Using the Pythagorean theorem:
   \[
   R^2 = (\sqrt{5})^2 + (R - r)^2
   \]
   Simplifying:
   \[
   R^2 = 5 + (R - r)^2
   \]
   Expanding the square:
   \[
   R^2 = 5 + (R^2 - 2Rr + r^2)
   \]
   Canceling \( R^2 \) on both sides:
   \[
   0 = 5 - 2Rr + r^2
   \]
   Rearranging the equation:
   \[
   2Rr = r^2 + 5
   \]
   This equation will help us relate the radii of the two circles.

4. **Area of the shaded region**:
   The area of the shaded region is the difference between the areas of the outer circle and the inner circle:
   \[
   \text{Area of shaded region} = \pi R^2 - \pi r^2 = \pi (R^2 - r^2)
   \]
   We can now solve for \( R \) and \( r \) and use this formula to compute the area.

Let me now calculate the solution for you.The expression for the area of the shaded region is:

\[
\text{Area of shaded region} = \pi \left( \frac{(r^2 + 5)^2}{4r^2} - r^2 \right)
\]

This formula allows us to calculate the shaded area once we know the radius \( r \) of the inner circle. 

Would you like to proceed with finding the radius \( r \), or do you have any further questions?

Here are 5 related questions:
1. How do we estimate the radius of the inner circle \( r \)?
2. How does the position of the chord affect the calculation of the shaded area?
3. Can this method be used for a non-circular inner region?
4. How would the calculation change if the chord wasn't tangent to the inner circle?
5. What geometric principles underlie the use of the Pythagorean theorem here?

**Tip**: The Pythagorean theorem is a powerful tool in circle geometry, especially when dealing with chords and tangents!

A chord of length √20 cm is tangential to the small circle in the figure below. Find the area of the shaded region, in cm².

Learn how to calculate the area of the shaded region between two circles when a chord is tangent to the inner circle. The problem involves the Pythagorean theorem and circle area formulas. Suitable for high school students (Grades 10-12).

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