Find the radius of a circle, ifthe end points of diameter of the circle are (2, 4) and (-3, -1).

Options

A. `3sqrt2` units
B. `5sqrt2`units
C. `(5sqrt2)/3` units
D. `(5sqr2)/2` units

Answer

To find the radius of a circle given the endpoints of its diameter, we can follow these steps:

### Step-by-Step Solution:

1. **Identify the endpoints of the diameter**:
The endpoints of the diameter are given as \( A(2, 4) \) and \( B(-3, -1) \).

2. **Use the distance formula to find the length of the diameter**:
The distance \( d \) between two points \( (x_1, y_1) \) and \( (x_2, y_2) \) is calculated using the formula:
\[ d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2} \]
Here, we can assign:
- \( x_1 = 2, y_1 = 4 \)
- \( x_2 = -3, y_2 = -1 \)

3. **Substitute the coordinates into the distance formula**:
\[ d = \sqrt{((-3) - 2)^2 + ((-1) - 4)^2} \]
\[ = \sqrt{(-5)^2 + (-5)^2} \]
\[ = \sqrt{25 + 25} \]
\[ = \sqrt{50} \]

4. **Calculate the diameter**:
The length of the diameter \( AB \) is \( \sqrt{50} \).

5. **Find the radius**:
The radius \( r \) is half of the diameter:
\[ r = \frac{d}{2} = \frac{\sqrt{50}}{2} \]

6. **Simplify the radius**:
We can simplify \( \sqrt{50} \):
\[ \sqrt{50} = \sqrt{25 \times 2} = 5\sqrt{2} \]
Thus, the radius becomes:
\[ r = \frac{5\sqrt{2}}{2} \]

### Final Answer:
The radius of the circle is \( \frac{5\sqrt{2}}{2} \) units.

---