Question

If the plane \( 56x + 4y + 9z = 2016 \) meets the coordinate axes in \( A, B \) and \( C \), then the centroid of the \( \triangle ABC \) is

• A. \( (12, 168, 224) \)
• B. \( (12, 168, 112) \)
• C. \( \left( 12, 168, \frac{224}{3} \right) \)
• D. \( \left( 12, -168, \frac{224}{3} \right) \)

Hint:

To find the points where a plane intersects the coordinate axes, set the other two coordinates to zero in the equation of the plane. These points are the vertices of the triangle. 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 \( \left( \frac{x_1 + x_2 + x_3}{3}, \frac{y_1 + y_2 + y_3}{3}, \frac{z_1 + z_2 + z_3}{3} \right) \).

Answer 1

The Correct Option is C

The plane intersects the x-axis when \( y = 0 \) and \( z = 0 \): \( 56x = 2016 \implies x = \frac{2016}{56} = 36 \).
So, \( A = (36, 0, 0) \).
The plane intersects the y-axis when \( x = 0 \) and \( z = 0 \): \( 4y = 2016 \implies y = \frac{2016}{4} = 504 \).
So, \( B = (0, 504, 0) \).
The plane intersects the z-axis when \( x = 0 \) and \( y = 0 \): \( 9z = 2016 \implies z = \frac{2016}{9} = 224 \).
So, \( C = (0, 0, 224) \).
The centroid of the triangle \( ABC \) with vertices \( (x_1, y_1, z_1), (x_2, y_2, z_2), (x_3, y_3, z_3) \) is given by \( \left( \frac{x_1 + x_2 + x_3}{3}, \frac{y_1 + y_2 + y_3}{3}, \frac{z_1 + z_2 + z_3}{3} \right) \).
Centroid of \( \triangle ABC = \left( \frac{36 + 0 + 0}{3}, \frac{0 + 504 + 0}{3}, \frac{0 + 0 + 224}{3} \right) \) Centroid \( = \left( \frac{36}{3}, \frac{504}{3}, \frac{224}{3} \right) = \left( 12, 168, \frac{224}{3} \right) \).