Question

In \(\triangle ABC\), \(O\) is the circumcenter and \(G\) is the centroid, then \[ OG^2= \] is:

• A. \(R^2-\dfrac{1}{3}(a^2+b^2+c^2)\)
• B. \(R^2-\dfrac{1}{6}(a^2+b^2+c^2)\)
• C. \(R^2-\dfrac{1}{9}(a^2+b^2+c^2)\)
• D. \(R-\dfrac{1}{9}(a^2+b^2+c^2)\)

Hint:

For a triangle, the distance between circumcenter \(O\) and centroid \(G\) is given by \[ OG^2=R^2-\frac{1}{9}(a^2+b^2+c^2) \] This is a standard result from triangle geometry.

Answer 1

The Correct Option is C

Step 1: Recall the standard formula.
In a triangle, if \(O\) is the circumcenter and \(G\) is the centroid, then the distance between them is given by \[ OG^2=R^2-\frac{1}{9}(a^2+b^2+c^2) \] Here, \(R\) is the circumradius and \(a,b,c\) are the side lengths of the triangle.

Step 2: Compare with the given options.
The formula \[ OG^2=R^2-\frac{1}{9}(a^2+b^2+c^2) \] matches option (3).

Step 3: Final conclusion.
Therefore, \[ \boxed{R^2-\frac{1}{9}(a^2+b^2+c^2)} \]