If the radisu of the circle passing through the origin and touching the line `x+y=2` at `(1, 1)` is r units, then the value of `3sqrt2r` is

Answer

To solve the problem, we need to find the radius \( r \) of the circle that passes through the origin (0, 0) and touches the line \( x + y = 2 \) at the point (1, 1). We will then calculate \( 3\sqrt{2}r \).

### Step-by-Step Solution:

1. **Identify the center of the circle**:
Since the circle touches the line \( x + y = 2 \) at the point (1, 1), the center of the circle must lie on the line that is perpendicular to \( x + y = 2 \) at (1, 1). The slope of the line \( x + y = 2 \) is -1, so the slope of the perpendicular line is 1. Therefore, the equation of the line through (1, 1) with slope 1 is:
\[ y - 1 = 1(x - 1) \implies y = x \]

2. **Find the coordinates of the center**:
Let the center of the circle be \( (h, h) \) since it lies on the line \( y = x \). The distance from the center \( (h, h) \) to the point of tangency \( (1, 1) \) is equal to the radius \( r \):
\[ r = \sqrt{(h - 1)^2 + (h - 1)^2} = \sqrt{2(h - 1)^2} = \sqrt{2}|h - 1| \]

3. **Find the distance from the center to the origin**:
The circle also passes through the origin (0, 0), so the distance from the center \( (h, h) \) to the origin must also equal the radius \( r \):
\[ r = \sqrt{h^2 + h^2} = \sqrt{2h^2} = |h|\sqrt{2} \]

4. **Set the two expressions for \( r \) equal**:
We have two expressions for \( r \):
\[ \sqrt{2}|h - 1| = |h|\sqrt{2} \]
Dividing both sides by \( \sqrt{2} \):
\[ |h - 1| = |h| \]

5. **Solve the absolute value equation**:
This gives us two cases:
- Case 1: \( h - 1 = h \) (not possible)
- Case 2: \( h - 1 = -h \) which simplifies to:
\[ 2h = 1 \implies h = \frac{1}{2} \]

6. **Find the radius \( r \)**:
Substitute \( h = \frac{1}{2} \) back into the expression for \( r \):
\[ r = |h|\sqrt{2} = \frac{1}{2}\sqrt{2} = \frac{\sqrt{2}}{2} \]

7. **Calculate \( 3\sqrt{2}r \)**:
Now we can find \( 3\sqrt{2}r \):
\[ 3\sqrt{2}r = 3\sqrt{2} \cdot \frac{\sqrt{2}}{2} = 3 \cdot \frac{2}{2} = 3 \]

### Final Answer:
Thus, the value of \( 3\sqrt{2}r \) is \( \boxed{3} \).