SOLUTION: The area of triangle ABC is trisected by two segments parallel to the base BC. The longer of these segments is 24 cm. What is the length of base BC? Express the answer in simplest

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Question 1114431: The area of triangle ABC is trisected by two segments parallel to the base BC. The longer of these segments is 24 cm. What is the length of base BC? Express the answer in simplest radical form
Answer by greenestamps(13367) About Me  (Show Source):
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Let the triangle be ABC, with shorter segment DE and longer segment FG parallel to BC, dividing the triangle into three regions of equal area.

Then triangles ADE, AFG, and ABC are all similar.

Furthermore, the area of triangle ADE is 1/2 the area of triangle AFG; and it is 1/3 the area of triangle ABC.

It is given that the length of FG is 24. Since the ratio of the areas of triangles ADE and AFG is 1%3A2, the ratio of side lengths between the two triangles is 1%3Asqrt%282%29. So the length of DE is 24%2Fsqrt%282%29+=+12%2Asqrt%282%29.

The ratio of the areas of triangles AFG and ABC is 1%3A3, so the ratio of lengths of sides between those two triangles is 1%3Asqrt%283%29. Since the length of FG is 12%2Asqrt%282%29, the length of BC is %2812%2Asqrt%282%29%29%2Asqrt%283%29+=+12%2Asqrt%286%29.

Answer: The length of base BC is 12%2Asqrt%286%29.