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>

Guest Apr 12, 2019

#1**+3 **

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

CPhill Apr 13, 2019