# Show by construction that the segment that connects the midpoint of two sides of triangle #DeltaABC# is parallel to the third and half it's length?

See the Proof in Explanation.

Hence, the Proof.

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To show that the segment connecting the midpoint of two sides of triangle ( \Delta ABC ) is parallel to the third side and half its length, follow these steps:

- Draw triangle ( \Delta ABC ).
- Mark the midpoints of two sides of the triangle. Let's call them ( M ) and ( N ), where ( M ) is the midpoint of side ( AB ) and ( N ) is the midpoint of side ( AC ).
- Connect ( M ) and ( N ) to form segment ( MN ).
- Draw a line parallel to side ( BC ) passing through point ( M ). Let this line intersect side ( AC ) at point ( P ).
- Since ( MN ) is parallel to side ( BC ), angle ( AMP ) and angle ( ABC ) are corresponding angles and are therefore equal.
- Also, since ( M ) is the midpoint of side ( AB ), ( AM = \frac{1}{2} AB ).
- Similarly, since ( N ) is the midpoint of side ( AC ), ( AN = \frac{1}{2} AC ).
- Using the midpoint theorem, segment ( MN ) is parallel to side ( BC ) and ( MN = \frac{1}{2} BC ).

Therefore, segment ( MN ) is parallel to side ( BC ) and half its length.

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