Consider the following statements
I. If G is the centroid of `DeltaABC`, then
GA = GB = GC.II. If H is the orthocentre of `Delta ABC`, then HA = HB = HC.
Which of the statements given above is/are correct ?

Options

A. Only I
B. Only II
C. Both I and II
D. Neither I nor II

Answer

To determine the correctness of the given statements about the centroid (G) and orthocenter (H) of triangle ABC, we will analyze each statement step by step.

### Step 1: Analyze Statement I
**Statement I:** If G is the centroid of triangle ABC, then GA = GB = GC.

1. **Definition of Centroid:** The centroid of a triangle is the point where the three medians intersect. It divides each median into two segments, one of which is twice the length of the other.

2. **Medians:** The centroid does not imply that the segments from the centroid to the vertices (GA, GB, GC) are equal. Instead, it states that the centroid divides each median in a 2:1 ratio.

3. **Conclusion for Statement I:** Since GA, GB, and GC are not necessarily equal (they are only equal in an equilateral triangle), this statement is **incorrect**.

### Step 2: Analyze Statement II
**Statement II:** If H is the orthocenter of triangle ABC, then HA = HB = HC.

1. **Definition of Orthocenter:** The orthocenter is the point where the three altitudes of the triangle intersect.

2. **Properties of Orthocenter:** Similar to the centroid, the distances from the orthocenter to the vertices (HA, HB, HC) are not necessarily equal. The orthocenter's distances to the vertices depend on the angles of the triangle.

3. **Conclusion for Statement II:** Since HA, HB, and HC are not necessarily equal (they are only equal in an equilateral triangle), this statement is also **incorrect**.

### Final Conclusion
Both statements I and II are incorrect. Therefore, the answer is that neither of the statements is correct.

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