Question

1. triangle def is formed by connecting the midpoints of the sides of triangle abc. the lengths of the sides of def are shown. what is the length of bc? (lesson 3 - 5)

a 3 units c 6 units
b 4 units d 8 units

1. if ab is 12, what is the length of de?

Explanation:

Step1: Recall the mid - segment theorem

The mid - segment of a triangle is parallel to the third side and half its length.

Step2: Analyze the relationship between \(\triangle DEF\) and \(\triangle ABC\)

In \(\triangle ABC\), \(D\), \(E\), \(F\) are mid - points. \(DE\), \(EF\), \(FD\) are mid - segments.
The side \(BC\) is related to the mid - segment \(DF\). By the mid - segment theorem, \(BC = 2\times DE\) (if we consider the mid - segment \(DE\) parallel to \(BC\)). But if we note that in a triangle formed by connecting mid - points (\(\triangle DEF\)), the sides of the larger triangle (\(\triangle ABC\)) are twice the sides of the smaller triangle (\(\triangle DEF\)).
Since the sides of \(\triangle DEF\) are \(2\), \(3\), \(4\) and we want to find \(BC\). If we assume the correct mid - segment relation (by the property of the mid - point triangle), \(BC\) corresponds to the side such that \(BC = 2\times3\) (because the side of \(\triangle DEF\) which is parallel to \(BC\) has length \(3\))

Step3: Calculate the length of \(BC\)

\(BC=6\) units.

Answer:

C. 6 units