: Hi helper, thanks for the help, this website is very good and it would not of been possible without your assistance.

Question

Here is a question i would like you to solve for me this time

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let ABC be a triangle. Two straight lines which are parallel to the side AC divide the triangle into three figures of equal are. In what parts is the side AB divided by the lines, if AB= 10 cm

Answer 1

Let the parallel lines DD' and EE' cut BA at D,E and BC at D', E'. Thus the sides BA and BC are now named BDEA and BD'E'C.

The question requires that the figures BDD', DD'E'E, and EE'CA have the same area. Thus the area of the triangles BDD', BEE' and BAC are in the ratio of 1, 2 and 3.

The area of similar triangles are proportional to the square of the sides, so
(BD/BA)²:(BE/BA)²:(BA/BA)²
=(1/3):(2/3):(3/3)

Therefore
BD=BA sqrt(1/3)
BE=BA sqrt(2/3)