Question

1. explain how you know that segment de is not parallel to segment bc.

Explanation:

Step1: Apply the converse of the basic proportionality theorem (Thales' theorem)

The converse of the basic proportionality theorem states that if a line divides two sides of a triangle proportionally, then the line is parallel to the third side. For \(\triangle ABC\) with line \(DE\), we need to check if \(\frac{AD}{DB}=\frac{AE}{EC}\).
Given \(AD = 4\), \(DB=2\), \(AE = 9\), and \(EC = 5\).
Calculate \(\frac{AD}{DB}=\frac{4}{2}=2\) and \(\frac{AE}{EC}=\frac{9}{5}=1.8\).

Step2: Compare the ratios

Since \(\frac{AD}{DB}
eq\frac{AE}{EC}\) (because \(2
eq1.8\)), by the converse of the basic proportionality theorem, segment \(DE\) is not parallel to segment \(BC\).

Answer:

Since \(\frac{AD}{DB}=\frac{4}{2} = 2\) and \(\frac{AE}{EC}=\frac{9}{5}=1.8\), and \(2
eq1.8\), by the converse of the basic proportionality theorem (Thales' theorem), segment \(DE\) is not parallel to segment \(BC\).