Draw a triangle and one of its medians. Then, find the area of each of the two triangles formed. Are they equal? If so, you could balance the triangle along that median.
See solution.
Practice makes perfect
Let's consider any triangle, and draw the median and altitude from the same vertex.
Now let's write the area of △ ABD and △ ADC.
A_(△ ABD) = BD* AH/2
A_(△ ADC) = DC* AH/2
Since D is the midpoint of BC, we have that BD = DC.
A_(△ ABD) = DC* AH/2
A_(△ ADC) = DC* AH/2
↘ ↙
A_(△ ABD) = A_(△ ADC)
Therefore, the median AD divides △ ABC into two triangles with the same area, so △ ABC can be balanced along that median. Additionally, the same situation happens with the remaining two medians.
Consequently, if we want to balance a triangle on one point, we need to find the intersection point of the three medians which is the centroid. This is why the centroid is the center of gravity of a triangle.
Balancing Point for a Rectangle
As we saw above, to find the balancing point or a rectangle we first draw the lines that divide it into rectangles with the same area.
From the graph above, each segment connecting the midpoints of opposite sides ( PQ and RS) divides rectangle ABCD into two rectangles with the same area. Therefore, the center of gravity of rectangle ABCD is the intersection point between PQ and RS.
