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

Options

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

Answer

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

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

### Step 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, \( (x_1, y_1) = (2, 4) \) and \( (x_2, y_2) = (-3, -1) \).

### Step 3: Substitute the coordinates into the distance formula
Substituting the coordinates into the formula, we get:
\[ d = \sqrt{((-3) - 2)^2 + ((-1) - 4)^2} \]
This simplifies to:
\[ d = \sqrt{(-5)^2 + (-5)^2} \]

### Step 4: Calculate the squares
Calculating the squares gives:
\[ d = \sqrt{25 + 25} = \sqrt{50} \]

### Step 5: Simplify the square root
The square root of 50 can be simplified:
\[ \sqrt{50} = \sqrt{25 \times 2} = 5\sqrt{2} \]
Thus, the length of the diameter \( d \) is \( 5\sqrt{2} \).

### Step 6: Find the radius
The radius \( r \) of the circle is half the diameter:
\[ r = \frac{d}{2} = \frac{5\sqrt{2}}{2} \]

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

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