Question

If the centroid of triangle whose vertices are \[ (a,1,3),\ (-2,b,-5)\ \text{and}\ (4,7,c) \] be the origin, then \[ a^2+b^2+c^2= \] is:

• A. \(68\)
• B. \(64\)
• C. \(72\)
• D. \(54\)

Hint:

The centroid of a triangle in three dimensions is obtained by taking the average of corresponding coordinates of the vertices.

Answer 1

The Correct Option is C

Step 1: Use the centroid formula.
The centroid of a triangle with vertices \[ (x_1,y_1,z_1),\ (x_2,y_2,z_2),\ (x_3,y_3,z_3) \] is \[ \left( \frac{x_1+x_2+x_3}{3}, \frac{y_1+y_2+y_3}{3}, \frac{z_1+z_2+z_3}{3} \right) \] Given that the centroid is the origin, \[ (0,0,0) \]

Step 2: Form equations using coordinates.
For the \(x\)-coordinate, \[ \frac{a+(-2)+4}{3}=0 \] \[ a+2=0 \] \[ a=-2 \] For the \(y\)-coordinate, \[ \frac{1+b+7}{3}=0 \] \[ b+8=0 \] \[ b=-8 \] For the \(z\)-coordinate, \[ \frac{3+(-5)+c}{3}=0 \] \[ c-2=0 \] \[ c=2 \]

Step 3: Find \[ a^2+b^2+c^2 \] Substitute the values: \[ a^2+b^2+c^2 = (-2)^2+(-8)^2+(2)^2 \] \[ =4+64+4 \] \[ =72 \]

Step 4: Final conclusion.
Hence, \[ \boxed{72} \]