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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

Triangles
Questioner:   Thomas
Country:  Australia
Category:  Advanced Math
Private:  No
 
Subject:  Similarity Problem
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.
....................................
Hi, Thomas,

Yes, it certainly does have something to do with similarity.  In the attached figure, there are three triangles, black, purple, green and they are similar. (Yuk, I could have chosen better colors, but..)

There is this number, called the 'ratio of similitude' (r-s) that gives the ratios of corresponding sides.  For example, if two triangles have r-s = 3, then each side of the big one is 3 times the corr. side of the little one.

AND THE AREAS ARE IN THE RATIO = the square of this r-s.  So if r-s = 3, the big one is 9 times the area.

Now in the picture:

Triangle Black = Area I
Purple = Area I + II = 2*Black.
Green = Area I + II + III = 3*Black.

Now the three triangles are similar, with 'ratio of similitude' (r-s) equal to:

Purple side = sqrt(2) * Black side.  
That means, each side of Purple = sqrt(2) * corresponding side of Black.

Green side = sqrt(3) * Black side.
That means, each side of Green = sqrt(3) * corresponding side of Black.

So AD/AC = sqrt(2)  and AB/AC = sqrt(3)

And we have AB = 10.  So I think now you can finish up.

Paul Klarreich

Expertise

I can answer questions in basic to advanced algebra (theory of equations, complex numbers), precalculus (functions, graphs, exponential, logarithmic, and trigonometric functions and identities), basic probability, and finite mathematics, including mathematical induction. I can also try (but not guarantee) to answer questions on Abstract Algebra -- groups, rings, etc. and Analysis -- sequences, limits, continuity. I won't understand specialized engineering or business jargon.

Experience

I taught at a two-year college for 25 years, including all subjects from algebra to third-semester calculus.

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