In `DeltaABC, A=(1,2), B=(5, 5), angleACB=90^(0)`. If area of `DeltaABC` is to be 6.5 sq. units, then the possible number of points for C is

Answer

To solve the problem, we need to find the possible number of points for point C in triangle ABC, given that A = (1, 2), B = (5, 5), and angle ACB = 90°. The area of triangle ABC is given as 6.5 square units.

### Step-by-Step Solution:

1. **Identify Points A and B**:
- A = (1, 2)
- B = (5, 5)

2. **Calculate the Length of AB**:
- Use the distance formula:
\[ AB = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2} \]
- Substituting the coordinates:
\[ AB = \sqrt{(5 - 1)^2 + (5 - 2)^2} = \sqrt{(4)^2 + (3)^2} = \sqrt{16 + 9} = \sqrt{25} = 5 \]

3. **Determine the Circumcircle**:
- Since angle ACB is 90°, point C must lie on the circumcircle of triangle ABC, where AB is the diameter.
- The radius of the circumcircle is half of AB:
\[ \text{Radius} = \frac{AB}{2} = \frac{5}{2} = 2.5 \]

4. **Area of Triangle Formula**:
- The area of triangle ABC can be calculated using the formula:
\[ \text{Area} = \frac{1}{2} \times \text{base} \times \text{height} \]
- Here, we can consider AB as the base and the height from point C to line AB.

5. **Set Up the Area Equation**:
- Given that the area is 6.5 square units, we can set up the equation:
\[ 6.5 = \frac{1}{2} \times AB \times h \]
- Substituting AB = 5:
\[ 6.5 = \frac{1}{2} \times 5 \times h \implies 6.5 = \frac{5h}{2} \implies 13 = 5h \implies h = \frac{13}{5} = 2.6 \]

6. **Compare Height with Radius**:
- The height (h) from point C to line AB must be equal to the radius of the circumcircle for point C to lie on the circle.
- The radius is 2.5, but we found h = 2.6, which is greater than the radius.

7. **Conclusion**:
- Since the height (2.6) is greater than the radius (2.5), point C cannot lie on the circumcircle of triangle ABC while satisfying the area condition.
- Therefore, there are **zero possible points for C**.

### Final Answer:
The possible number of points for C is **0**.