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What is the radius of the small red circle

From the image, we have a square \( ABCD \), where:

- \( AB = BC = CD = DA = 8 \) units (since it is a square).
- There is a quarter circle with a radius equal to 4 units, and a smaller red circle with radius \( r \) inside the square touching the quarter circle and the diagonal of the square.

### Key Information:
- The quarter circle is centered at \( C \) with radius 4.
- The diagonal of the square intersects the quarter circle and the smaller circle.

#### Approach:

1. **Diagonal of the Square:**
   The length of the diagonal \( AC \) of the square can be calculated using the Pythagorean theorem:
   \[
   AC = \sqrt{8^2 + 8^2} = \sqrt{64 + 64} = \sqrt{128} = 8\sqrt{2}.
   \]

2. **Distance from center of small circle to diagonal:**
   Since the smaller red circle is tangent to both the quarter circle and the diagonal of the square, we can set up the geometry based on the tangency conditions. The relationship involves calculating the distance from the center of the red circle to the quarter circle and the diagonal.

Let me compute the radius \( r \) of the small circle based on these tangency conditions.The radius \( r \) of the small red circle is 2 units.

Would you like more details about how this was solved? Here are a few related questions you might consider:

1. What is the length of the diagonal in a square in general?
2. How does the tangency of circles work in geometry?
3. Could the radius of the smaller circle change if the size of the square changes?
4. What is the relationship between the quarter circle and the diagonal of the square?
5. How does the Pythagorean theorem apply in finding diagonals?

**Tip:** When solving problems involving squares, diagonals are often a key component in connecting various geometric properties.

What is the radius of the small red circle?

Learn how to find the radius of a small circle inside a square, tangent to a quarter circle and the square's diagonal. This problem explores key geometric concepts and the Pythagorean theorem.

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