Multiple choice

The distance from the origin to the centroid of the tetrahedron formed by the points $(0, 0, 0), (a, 0, 0), (0, b, 0), (0, 0, c)$ is:

  1. $\displaystyle \dfrac{\sqrt{a+{b}+{c}}}{4}$
  2. $\displaystyle \dfrac{\sqrt{{a}+{b}+{c}}}{3}$
  3. $\displaystyle \dfrac{\sqrt{{a}^{2}+{b}^{2}+{c}^{2}}}{16}$
  4. $\displaystyle \dfrac{\sqrt{{a}^{2}+{b}^{2}+{c}^{2}}}{4}$
Reveal answer Fill a bubble to check yourself
D Correct answer
Explanation

The centroid of a tetrahedron with vertices (x1,y1,z1), (x2,y2,z2), (x3,y3,z3), (x4,y4,z4) is ((x1+x2+x3+x4)/4, (y1+y2+y3+y4)/4, (z1+z2+z3+z4)/4). Here, vertices are (0,0,0), (a,0,0), (0,b,0), (0,0,c). Centroid = (a/4, b/4, c/4). Distance from origin = sqrt((a/4)^2 + (b/4)^2 + (c/4)^2) = sqrt(a^2 + b^2 + c^2) / 4.

AI explanation

The centroid of a tetrahedron is found by averaging the coordinates of its four vertices. Adding the coordinates of (0, 0, 0), (a, 0, 0), (0, b, 0), and (0, 0, c) gives the centroid as (a/4, b/4, c/4). The distance from the origin (0, 0, 0) to this centroid is calculated using the 3D distance formula, resulting in the square root of ((a/4) squared + (b/4) squared + (c/4) squared). This simplifies to the square root of (a squared + b squared + c squared) divided by 4.

Keep practicing — related questions