Question

prove that in any triangle the line segment joining the midpoints of any two sides is parallel to the third side and is half of it​

Answer 1

Given :-

Let us consider

A triangle ABC such that

• D is the midpoint of AB

and

• E is the midpoint of AC.

To Prove :-

$\rm :\longmapsto\:DE = \dfrac{1}{2} BC \: and \: DE \: \parallel \: BC$

Construction :-

• Through C, draw a line CF parallel to AB, intersects DE at F when produced.

So,

• CF || DB

Proof :-

Since, it is given that

In triangle ABC

• D is the midpoint of AB.

• ⇛ AD = DB

Also,

• E is the midpoint of AC

• ⇛ AE = EC

Now,

$\rm :\longmapsto\:In \: \triangle \: EAD \: and \: \triangle \: ECF$

$\rm :\longmapsto\:AE = EC \: \: \: \: \: \: \{ \red{ \bf \: given} \}$

$\rm :\longmapsto\: \angle \: AED = \angle \: CEF \: \: \: \: \: \{ \red{ \bf \: v.o. \: angles} \}$

$\rm :\longmapsto\: \angle \: EAD = \angle \: ECF \: \: \: \: \: \{ \red{ \bf \: alternate \: angles} \}$

$\bf\implies \:\triangle \: EAD \: \cong \: \triangle \: ECF \: \: \: \: \: \{ \red{ \bf \: ASA }\}$

$\rm :\implies\:DE = EF \: \: \: \: \: \: \{CPCT \} - - (1)$

and

$\rm :\longmapsto\:AD = FC \: \: \: \: \{CPCT \}$

But,

$\rm :\longmapsto\:AD = DB \: \: \: \: \: \{given \}$

$\rm :\implies\:AD = DB= FC$

Now,

• In quadrilateral DBCF,

we have,

$\rm :\longmapsto\:DB= FC \: \: \: and \: \: \: DB \: \parallel \: FC$

• We know, if in a quadrilateral, one pair of opposite sides are parallel and equal, then quadrilateral is a parallelogram.

$\rm :\implies\:DBCF \: is \: a \: parallelogram.$

$\rm :\longmapsto\: \: \: \therefore \: DF \: = \: BC \: \: \: \: \: and \: \: \: \: DF \: \parallel \: BC$

$\rm :\longmapsto\: \: \: 2DE \: = \: BC \: \: \: \: \: and \: \: \: \: DE \: \parallel \: BC$

$\rm :\implies\: \: \: DE \: = \: \dfrac{1}{2} \: BC \: \: \: \: \: and \: \: \: \: DE \: \parallel \: BC$

${\boxed{\boxed{\bf{Hence, Proved}}}}$

Note :-

The given statement is called Midpoint Theorem.

Additional Information :-

Its Converse states that

If a line is drawn through the midpoint of one side of a triangle, and parallel to the other side, it bisects the third side