Prove that “The line segment joining the mid-points of two sides of a triangle is parallel to thethird side and half of it.”
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Proof: Consider triangle ABC.
Join mid-points of AB and AC.
Let mid-point of AB be D and mid-point of BC be E.
Now we have two triangles ABC and ADE
In triangle ABC and triangle ADE
AB/AD=AC/AE=2------- (by construction)
angle BAC=angle DAE------ (as A-D-B and A-E-C.)
thus triangle ABC∼ triangle ADE
Thus angle ADE = angle ABC as triangles are similar
and DE ∥ BC as corresponding angles are equal.
Also DE/BC=AD/AB=1/2 (sides of similar triangle.)
Join mid-points of AB and AC.
Let mid-point of AB be D and mid-point of BC be E.
Now we have two triangles ABC and ADE
In triangle ABC and triangle ADE
AB/AD=AC/AE=2------- (by construction)
angle BAC=angle DAE------ (as A-D-B and A-E-C.)
thus triangle ABC∼ triangle ADE
Thus angle ADE = angle ABC as triangles are similar
and DE ∥ BC as corresponding angles are equal.
Also DE/BC=AD/AB=1/2 (sides of similar triangle.)
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