Mathematics

Mathematics

Problem

If DD and EE are the midpoints of sides ABAB and ACAC of △ABC\triangle ABC, then which of the following is always true? a) DE=12 BC and DE∥ACDE = \dfrac{1}{2}\,BC \text{ and } DE \parallel AC b) DE=12 BC and DE∥BCDE = \dfrac{1}{2}\,BC \text{ and } DE \parallel BC c) DE=BC and DE∥ACDE = BC \text{ and } DE \parallel AC d) DE=12 AB and DE∥ACDE = \dfrac{1}{2}\,AB \text{ and } DE \parallel AC
Intuitive explanation

The key insight is the midsegment theorem: when you connect the midpoints of two sides of a triangle, you create a segment that's parallel to the third side and exactly half its length. Rather than getting lost in coordinate calculations, recognize that DEDE has a special relationship to BCBC—it's not just any segment, but one with guaranteed geometric properties that hold for any triangle you choose.

Intuitive explanation

The key insight is the midsegment theorem: when you connect the midpoints of two sides of a triangle, you create a segment that's parallel to the third side and exactly half its length. Rather than getting lost in coordinate calculations, recognize that DEDE has a special relationship to BCBC—it's not just any segment, but one with guaranteed geometric properties that hold for any triangle you choose.

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