Question

Let the circle with centre at origin pass through the vertices of an equilateral triangle ABC . If \(A \equiv(2,4)\), then the length of the median through A is

• A. \(2\sqrt{5}\) units
• B. \(3\sqrt{5}\) units
• C. \(4\sqrt{5}\) units
• D. \(6\sqrt{5}\) units

Hint:

In an equilateral triangle the centroid and circumcentre coincide, so the median is 3/2 of the circumradius.

Answer 1

The Correct Option is B

Step 1: Understanding the Concept
In an equilateral triangle the circumcentre and centroid coincide, and the centroid divides each median in the ratio \(2:1\).

Step 2: Find the radius
The centre is the origin, so \(R=OA=\sqrt{2^2+4^2}=\sqrt{20}=2\sqrt5\).

Step 3: Calculation
The median from A has length \(OA+\tfrac12 OA=\tfrac32R\).
\(\tfrac32\times2\sqrt5=3\sqrt5\).

Final Answer:
The median through A has length \(3\sqrt5\) units. \[ \boxed{\text{(B) }3\sqrt{5}\text{ units}} \]