Question

A triangle is formed by the lines \[x + y = 8\], x-axis and y-axis. Find its centroid.
A. \[\left( {\dfrac{8}{3},\dfrac{8}{3}} \right)\]
B. \[\left( {8,8} \right)\]
C. \[\left( {4,4} \right)\]
D. \[\left( {0,0} \right)\]

Answer 1

Hint: First of all, find the point of intersection of the given three lines and draw the figure with the obtained points. Then find the centroid of these three points by using the centroid formula. So, use this concept to reach the solution of the given problem.

Given lines are \[x + y = 8\], x-axis and y-axis.

The line equation of x-axis is \[y = 0\]

The line equation of y-axis is \[x = 0\]

Solving the lines \[x + y = 8\] and \[y = 0\] we get

\[

  x + 0 = 8 \\

  \therefore x = 0 \\

\]

So, the point of intersection of the lines \[x + y = 8\] and \[y = 0\] is \[A\left( {8,0} \right)\]
Solving the lines \[x + y = 8\] and \[x = 0\] we get

\[

  0 + y = 8 \\

  \therefore y = 8 \\

\]

So, the point of intersection of the lines \[x + y = 8\] and \[x = 0\] is \[B\left( {0,8} \right)\]
And the point of intersection of the lines \[x = 0\] and \[y = 0\] is \[O\left( {0,0} \right)\]
The points formed are shown in the below figure:

We know that the centroid of a triangle with vertices \[{\text{A }}\left( {{x_1},{y_1}} \right)\], \[{\text{B }}\left( {{x_2},{y_2}} \right)\] and \[{\text{C }}\left( {{x_3},{y_3}} \right)\] is given by the formula \[{\text{G }}\left( {x,y} \right) = \left( {\dfrac{{{x_1} + {x_2} + {x_3}}}{3},\dfrac{{{y_1} + {y_1} + {y_1}}}{3}} \right)\].

So, the centroid formed by the given lines is

\[

  G = \left( {\dfrac{{8 + 0 + 0}}{3},\dfrac{{0 + 8 + 0}}{3}} \right) \\

  \therefore G = \left( {\dfrac{8}{3},\dfrac{8}{3}} \right) \\

\]

Thus, the correct option is A. \[\left( {\dfrac{8}{3},\dfrac{8}{3}} \right)\]

Note: The line equation of x-axis is \[y = 0\] and the line equation of y-axis is \[x = 0\]. The centroid of a triangle with vertices \[{\text{A }}\left( {{x_1},{y_1}} \right)\], \[{\text{B }}\left( {{x_2},{y_2}} \right)\] and \[{\text{C }}\left( {{x_3},{y_3}} \right)\] is given by the formula\[{\text{G }}\left( {x,y} \right) = \left( {\dfrac{{{x_1} + {x_2} + {x_3}}}{3},\dfrac{{{y_1} + {y_1} + {y_1}}}{3}} \right)\].