In a right-angled triangle ABC, the angle at B is 90°. If AB = 6 cm and BC = 8 cm, what is the distance between the centroid G an…

Question

In a right-angled triangle ABC, the angle at B is 90°. If AB = 6 cm and BC = 8 cm, what is the distance between the centroid G and the orthocenter H?

1. A. 8 cm
2. B. 5 cm
3. C. 10/3 cm (Correct answer)
4. D. 20/3 cm

Correct Answer

Option C — 10/3 cm

Detailed Solution & Explanation

The correct answer is 10/3 cm.

Key Points

• In a right-angled triangle the orthocentre is the right-angle vertex itself, so H is at B.
• Place B at the origin: B(0, 0), A(0, 6) and C(8, 0), since AB = 6 and BC = 8.
• The centroid is the average of the three vertices:
  • G = ((0 + 0 + 8)/3, (0 + 6 + 0)/3) = (8/3, 2)
• Distance GH = √((8/3)² + 2²) = √(64/9 + 36/9) = √(100/9) = 10/3 cm.

Additional Information

• The hypotenuse AC = √(6² + 8²) = 10 cm — the 6-8-10 triple — and the circumcentre of a right triangle is the midpoint of the hypotenuse, here (4, 3).
• The centroid always divides the segment joining the orthocentre and circumcentre in a 2 : 1 ratio, so GH = (2/3) × OH. Since OH = 5 here, GH = 10/3 ✓ — a faster route than coordinates.
• Those three points are collinear on the Euler line of any triangle.
• Placing the right angle at the origin turns a geometry problem into simple arithmetic, and is worth doing whenever a right angle is given.