Let the orthocentre and centroid of a triangle be `A(-3,5) and B(3,3)` respectively. If C is the circumcentre of the triangle then the radius of the circle having line segment AC as diameter, is

Options

A. 7.74
B. 6.74
C. 5.74
D. 4.74

Explanation

To find the radius of the circle having line segment AC as diameter, we will follow these steps:

### Step 1: Identify the coordinates of the orthocenter (A) and centroid (B)
Given:
- Orthocenter \( A(-3, 5) \)
- Centroid \( B(3, 3) \)

### Step 2: Use Euler's theorem to find the circumcenter (C)
According to Euler's theorem, the orthocenter (A), centroid (B), and circumcenter (C) of a triangle are collinear, and the centroid divides the segment joining the orthocenter and circumcenter in the ratio \( 2:1 \).

Let the coordinates of circumcenter \( C \) be \( (x, y) \).

Using the section formula for the x-coordinates:
\[ B_x = \frac{2C_x + A_x}{3} \]
Substituting the known values:
\[ 3 = \frac{2x - 3}{3} \]
Cross-multiplying gives:
\[ 9 = 2x - 3 \]
Solving for \( x \):
\[ 2x = 12 \implies x = 6 \]

Now, for the y-coordinates:
\[ B_y = \frac{2C_y + A_y}{3} \]
Substituting the known values:
\[ 3 = \frac{2y + 5}{3} \]
Cross-multiplying gives:
\[ 9 = 2y + 5 \]
Solving for \( y \):
\[ 2y = 4 \implies y = 2 \]

Thus, the coordinates of circumcenter \( C \) are \( (6, 2) \).

### Step 3: Calculate the length of segment AC
Now, we need to find the length of segment AC, where:
- \( A(-3, 5) \)
- \( C(6, 2) \)

Using the distance formula:
\[ AC = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2} \]
Substituting the coordinates:
\[ AC = \sqrt{(6 - (-3))^2 + (2 - 5)^2} \]
\[ = \sqrt{(6 + 3)^2 + (2 - 5)^2} \]
\[ = \sqrt{9^2 + (-3)^2} \]
\[ = \sqrt{81 + 9} = \sqrt{90} = 3\sqrt{10} \]

### Step 4: Find the radius of the circle
Since AC is the diameter of the circle, the radius \( r \) is half of AC:
\[ r = \frac{AC}{2} = \frac{3\sqrt{10}}{2} \]

### Final Answer
The radius of the circle having line segment AC as diameter is \( \frac{3\sqrt{10}}{2} \).

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