Question

Prove that the line segment joining the mid point of two sides of a triangle is parallel to the third side and half of it.

Answer 1

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Proof: Consider triangle ABC.
Join mid-points of AB and AC.
Let mid-point of AB be M and mid-point of BC be N.
Now we have two triangles ABC and AMN
In triangle ABC and triangle AMN
AB/AM=AC/AN=2..........by construction
angle BAC=angle MAN.........as A-M-B and A-N-C.
thus triangle ABC∼∼ triangle AMN.
Thus angle AMN=angle ABC as triangles are similar
and MN ∥∥ BC as corresponding angles are equal.
Also MN/BC=AM/AB=1/2 (sides of similar triangle.)