Given the line 12x + 9y = 36
the line intersects the x-axis when y = 0,
i.e. 12x = 36 ⇒ x = 3
Thus the x-intecept is (3, 0)
the line intersects the y-axis when x = 0,
i.e. 9y = 36 ⇒ y = 4
Thus, the y-intercept is (0, 4)
Thus, the coordinates of the vertices of the triangle are (0, 0), (0, 4) and (3, 0)
The midpoint of the side of the rectangle joining points (0, 0) and (0, 4) is (0, 2) and the equation of the median joining points (0, 2) and (3, 0) is given by
[tex] \frac{y-2}{x} = \frac{-2}{3} \\ \\ \Rightarrow3(y-2)=-2x \\ \\ \Rightarrow y=- \frac{2}{3} x+2[/tex]
The midpoint of the side of the rectangle joining points (0, 0) and (3, 0) is (1.5, 0) and the equation of the median joining points (1.5, 0) and
(0, 4) is given by
[tex] \frac{y}{x-1.5} = \frac{4}{-1.5} \\ \\ \Rightarrow1.5y=-4(x-1.5)=-4x+6 \\ \\ \Rightarrow y=- \frac{8}{3} x+4[/tex]
The point of intersetion of the two medians is given by
[tex]- \frac{2}{3} x+2=- \frac{8}{3} x+4 \\ \\ \Rightarrow2x=2 \\ \\ \Rightarrow x=1 \\ \\ \Rightarrow y=-\frac{2}{3}+2=\frac{4}{3}[/tex]
Therefore, the centroid of the given triangle is located at point [tex]\left(1, \frac{4}{3} \right)[/tex]
