AB is a line segment and M is its mid point. Semi-circles are drawn with AM, MB and AB as diameters on the same side of AB. A circle is drawn to touch all the three semi-circles. Its radius is :

Options

A. `(AB)/3`
B. `2/3 AB`
C. `(AB)/6`
D. `3/4 AB`

Explanation

To find the radius of the circle that touches all three semi-circles drawn on the line segment AB, we can follow these steps:

### Step-by-Step Solution:

1. **Define the Length of AB**:
Let the length of line segment AB be \( AB = 4x \). Since M is the midpoint of AB, we have:
\[ AM = MB = 2x \]

2. **Identify the Radii of the Semi-Circles**:
- The radius of the semi-circle with diameter AM is:
\[ r_1 = \frac{AM}{2} = \frac{2x}{2} = x \]
- The radius of the semi-circle with diameter MB is:
\[ r_2 = \frac{MB}{2} = \frac{2x}{2} = x \]
- The radius of the semi-circle with diameter AB is:
\[ r_3 = \frac{AB}{2} = \frac{4x}{2} = 2x \]

3. **Set Up the Geometry**:
Let the radius of the circle that touches all three semi-circles be \( r \). The center of this circle will be at a distance \( (2x - r) \) from point M (the midpoint of AB) and at a distance \( r \) from the other two semi-circles.

4. **Use the Right Triangle Relationship**:
We can form a right triangle with:
- One leg as the distance from the center of the circle to M, which is \( 2x - r \).
- The other leg as the distance from the center of the circle to the point where it touches the semi-circle with diameter AB, which is \( r \).
- The hypotenuse is the distance from the center of the circle to the center of the semi-circle with diameter AB, which is \( 2x \).

Using the Pythagorean theorem:
\[ (2x - r)^2 + r^2 = (2x)^2 \]

5. **Expand and Simplify the Equation**:
Expanding the left side:
\[ (2x - r)^2 = 4x^2 - 4xr + r^2 \]
Thus, the equation becomes:
\[ 4x^2 - 4xr + r^2 + r^2 = 4x^2 \]
Simplifying gives:
\[ 2r^2 - 4xr = 0 \]

6. **Factor the Equation**:
Factoring out \( 2r \):
\[ 2r(r - 2x) = 0 \]
This gives us two solutions:
\[ r = 0 \quad \text{or} \quad r = 2x \]

7. **Find the Radius in Terms of AB**:
Since \( r \) cannot be zero, we take \( r = 2x \). Recall that \( AB = 4x \), thus:
\[ r = \frac{AB}{2} \]

8. **Final Result**:
The radius of the circle that touches all three semi-circles is:
\[ r = \frac{AB}{6} \]