How to divide a triangle in 3 equal parts from altitude to hypotenuse?

Question

How can I divide a right angled triangle into 3 equal parts having equal areas using lines parallel to base from altitude to hypotenuse? Actually this is a piece of land and we want it to divide in such a way that.

 Area of a = Area of b = Area of c

Answer 1

See that when you would try to divide your triangle using lines parallel to the base , you will get three similar triangles.

Using the fact that for two similar triangles if their sides are in the ratio $\frac{p}{q} \ $ the areas are in ratio $\frac{p^2}{q^2}$.

Here since the areas a and a+b are in ratio $\frac{1}{2}\ $, it means that the sides would be in ratio $\frac{1}{\sqrt{2}}\ $, similarly sides of a and a+b+c would be in ratio $\frac{1}{\sqrt{3}} \ $.

which means you need to divide your altitude into three segments in ratio $1:\sqrt{2}-1:\sqrt{3}-\sqrt{2}$.