If I have a triangle with the coodinates A(0,0) B (30,0) and C (30,40)
Is the centre of gravity 1/3 of the base and 1/3 of the height?
would this make it (10, 40/3) ??
The center of gravity will balance the triangle in the x direction, as well as in the y direction.
The area of the triangle is
$\displaystyle \frac{30 \times 40}{2} = 600$
You now need a triangle such that $\displaystyle \frac{X \times Y}{2} = 300 $
The ratio of the main triangle's height to the base is $\displaystyle \frac {40}{30}$
You need to have $\displaystyle \frac{Y}{X} = \frac{4}{3}$ or $\displaystyle Y \, = \, 1.33333 X $
$\displaystyle \frac {X \times 1.33333 X}{2}=300$
$\displaystyle X^2 = 450 $
$\displaystyle X = \sqrt{450} = 21.2132 $
The center of gravity will occur along the triangle with the x coordinate equal to 21.2132.
You will need to do the appropriate calculations for the y coordinate.
You want to cut the large triangle so that the upper or smaller triangle is half the area of the larger triangle.