Problem

Use the converse of the side-splitter theorem to decide whether the line must be parallel to the third side: \(AD/DB=4/6\) and \(AE/EC=6/9\)

Neutral side-ratio reference stating AD over DB equals 4 over 6 and AE over EC equals 6 over 9, without a cutting segment or parallel relation.

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What this problem is really about

The converse side-splitter theorem turns proportional division into a parallel-line conclusion. We’ll form the two split-side ratios in matching order, simplify them and confirm with cross products, verify the converse hypothesis exactly, and then state the required relationship between the interior segment and the third side.

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Yes, because 4/6 and 6/9 both simplify to 2/3, the sides are divided proportionally, so DE is parallel to BC.

No, because 4/6 and 6/9 are the lengths of different sides, and the converse of the side-splitter theorem cannot be used here.

Yes, because the ratios match, which means the triangles are already similar, but that does not prove DE is parallel to BC.

No, because the fractions 4/6 and 6/9 use different numbers, so there is not enough evidence that DE is parallel to BC.

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Use the converse of the side-splitter theorem to decide whether the line must be parallel to the third side: \(AD/DB=4/6\) and \(AE/EC=6/9\)

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