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Thales' Theorem and the Theorem of Proportional Segments

] Difficulty [Ratio of Areas of Triangles with a Common Base or Common Height]

In triangle $ABC$, with an area of 6, a point $K$ is taken on side $AB$, dividing this side in the ratio $AK: BK = 2: 3$, and a point $L$ is taken on side $AC$, dividing $AC$ in the ratio $AL: LC = 5: 3$. The point $Q$, the intersection of lines $CK$ and $BL$, is 1.5 units away from line $AB$. Find the side $AB$.

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Draw a line through point $K$ parallel to $BL$, until it intersects side $AC$ at point $M$. By Thales' theorem, $LM = 3/5 AL = CL$. Therefore, $CQ = KQ$.

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