Geometry Thing

In Triangle LMN, altitude LK is 12cm long. Through point J of LK a line is drawn parallel to MN, dividing the triangle into two regions with equal areas. find LJ>

Note that  the area of the large triangle must be twice that of the smaller triangle

Call the area of the large triangle L   and the area of the smaller triangle , S

Then    2S = L 

So.....

area of smaller triangle * (scale factor)^2   = area of large triangle

S * (scale factor)^2  = 2S       divide both sides by S

scale factor^2  = 2   take the square root of both sides

scale factor = √2

This means that every dimension of the larger triangle is √2 that of the smaller triangle

Or....put another way.....every dimension of the smaller triangle is 1 /√2 that of the larger triangle

So....since LK  = 12 = altitude of larger triangle

Then ....the altitude of the smaller triangle, LJ  = 12 / √2   = 6√2 units

Check

Area of large triangle  = (1/2)MN * 12  = 6MN

Base of smaller triangle = MN/√2 

Height of smaller triangle  6√2

So  area of smaller triangle = (1/2) (MN/√2)(6√2)  =  (1/2)(6)MN  = 3 MN  = 1/2 area of larger triangle