In the (x, y) coordinate plane, what is the radius of the circle having the point `(2, -4) and (-4, 4)` as endpoints of a diameter?

Answer

To find the radius of the circle with endpoints of a diameter at the points \( (2, -4) \) and \( (-4, 4) \), we will follow these steps:

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

### 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:
\[ d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2} \]
Substituting the coordinates of points A and B into the formula:
- \( x_1 = 2, y_1 = -4 \)
- \( x_2 = -4, y_2 = 4 \)

Now, we calculate:
\[ d = \sqrt{((-4) - 2)^2 + (4 - (-4))^2} \]
\[ = \sqrt{(-6)^2 + (8)^2} \]
\[ = \sqrt{36 + 64} \]
\[ = \sqrt{100} \]
\[ = 10 \]

### Step 3: Calculate the radius of the circle
The radius \( r \) of the circle is half of the diameter:
\[ r = \frac{d}{2} = \frac{10}{2} = 5 \]

### Final Answer
The radius of the circle is \( 5 \) units.

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