Multiple choice

The vertices of a triangle are $(6, 6), (0, 6)$ and $(6, 0)$. The distance between its circumcentre and centroid is

  1. $2\sqrt 2$
  2. $2$
  3. $\sqrt 2$
  4. $1$
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C Correct answer
Explanation

Vertices are (6,6), (0,6), (6,0). This is a right triangle. Circumcentre is the midpoint of the hypotenuse: ((6+0)/2, (6+6)/2) = (3,6) is wrong, it's (3,3). Centroid is ((6+0+6)/3, (6+6+0)/3) = (4,4). Distance = sqrt((4-3)^2 + (4-3)^2) = sqrt(1+1) = sqrt(2).

AI explanation

The given points (6,6), (0,6), and (6,0) form a right-angled triangle because the lines along (0,6) to (6,6) and (6,6) to (6,0) are perpendicular. The circumcentre of a right-angled triangle is the midpoint of its hypotenuse, making it (3,3). The centroid is found by averaging the coordinates, giving (4,4). The distance between the circumcentre (3,3) and the centroid (4,4) is sqrt((4-3)^2 + (4-3)^2), which equals sqrt(2).

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