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Expert: - 7/12/2009


Question
Hi,
I've got a maths problem:
Let ABC be a triangle. Two straight lines which are parallel to the side AC divide the triangle into three figures of equal area. In what parts is the side AB divided by the lines, if AB = 10 cm?

I know it has something to do with similarity, but I really have no idea as to where to start the problem.

Thankyou.

Answer
Let C be the top of every triangle mentioned.
The area of any such triangle will be kl˛ where k is some unknown constant and l is the lenght down the side for C.

The area of the entire triangle, then is 100k.  
That means the area for each of them would be 100k/3.

Suppose that the triangle has been divided into equal portion.
This has been done with two line that are parallel to the bottom.
The 1st line is L1 from the top and the 2nd line is L2 from the top.

The area of the top triangle is k*L1˛.  Is has already been pointed out the area of that triangle was also 100k/3.  Setting these equal to each other gives k*L1˛ = 100k/3.  Both sides can be divided by k, giving L1˛ = 100/3.  Taking the squareroot of both sides gives
L1 = 10/√3.  That is how far to put the first line from the top.

Now lets look at the triangle formed by L2.  We know that the area of this triangle is k*L2˛.  We also know that the area is the middle area plus the top area.  These two areas are equal, each with 100k/3.
That means that k*L2˛ = 100k/3 + 100k/3.  Dividing by k gives us
L2˛ = 200/3.  What that says is that L2 = 10√2/√3.  That is how far to put the second line from the top.

This leaves 100 - L2 as the height of the third area,
so L2 = 10 - 10√2/√3 = 10√3/√3 - 10√2/3 = 10(√3-√2)/3.


Going back and checking these answers gives the to triangle has area of k*(10/√3)˛ = k*100/3.

The area of the trapezoid in the middle is the difference of the areas in the two triangles, one with bottom L2 and the other determined by L1.  That is, the value is k((10√2/√3)˛ - (10/√3)˛) =
k((200/3 - 100/3) = k*100/3.

The area of the last one would be the area of the entire triangle - the area of the triangle defined by L2.  That is,
k*10˛ - k*(100√2/√3)˛ = k(100 - 200/3) = k*100/3.

We have now found the lines and shown that all of the area are equal.

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