A triangle A B C is placed so that the mid-points of the sides are on the x , y , z axes. Lengths of the intercepts made by the plane containing the triangle on these axes are respectively \alpha , \beta , \gamma. Coordinates of the centroid of the triangle A B C are

Engineering

Mathematics

Line of Greatest Slope

Question

A triangle

A B C

is placed so that the mid-points of the sides are on the

x, y, z

axes. Lengths of the intercepts made by the plane containing the triangle on these axes are respectively

α, β, γ

. Coordinates of the centroid of the triangle

A B C

are

(- α / 3, β / 3, γ / 3)

(α / 3, - β / 3, γ / 3)

(α / 3, β / 3, - γ / 3)

(α / 3, β / 3, γ / 3)

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△ A B C

has

P

as mid-point of

A B

on x-axis.

Q

as midpoint of

B C

on y-axis.

R

as midpoint of

A C

on z-axis.

∴ A = (x_{1} y_{1} z_{1})

∴ B = (x_{2} y_{2} z_{2})

∴ C = (x_{3} y_{3} z_{3})

\Rightarrow P = (\frac{x_{1} + x_{2}}{2}, \frac{y_{1} + y_{2}}{2}, \frac{z_{1} + z_{2}}{2}) Q = (\frac{x_{2} + x_{3}}{2}, \frac{y_{2} + y_{3}}{2}, \frac{z_{2} + z_{3}}{2}) R = (\frac{x_{3} + x_{1}}{2}, \frac{y_{3} + y_{1}}{2}, \frac{z_{3} + z_{1}}{2}) P = (α, 0, 0) Q = (0, β, 0) R = (0, 0, γ) ∴ \frac{x_{1} + x_{2}}{2} = α, \frac{y_{1} + y_{2}}{2} = 0, \frac{z_{1} + z_{2}}{2} = 0 \frac{x_{2} + x_{3}}{2} = 0, \frac{y_{2} + y_{3}}{2} = β, \frac{z_{2} + z_{3}}{2} = 0 \frac{x_{3} + x_{1}}{2} = 0, \frac{y_{3} + y_{1}}{2} = 0, \frac{z_{3} + z_{1}}{2} = γ ∴ x_{1} + x_{2} + x_{3} = α, y_{1} + y_{2} + y_{3} = β, z_{1} + z_{2} + z_{3} = γ

∴

centroid

(G) = (\frac{x_{1} + x_{2} + x_{3}}{3}, \frac{y_{1} + y_{2} + y_{3}}{3}, \frac{z_{1} + z_{2} + z_{3}}{3}) = (\frac{α,}{3}, \frac{β,}{3}, \frac{γ,}{3})

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