Question

In AABC, D, E and F are the mid-points of the sides AB, BC and AC, respectively. Then, prove that quadrilateral DECF is a parallelogram.​

Answer 1

$\large\underline{\sf{Solution-}}$

Given that, in triangle ABC

• D is the midpoint of AB

• E is the midpoint of BC

• F is the midpoint of AC

Now, In triangle ABC,

• D is the midpoint of AB

• F is the midpoint of AC.

So, By midpoint theorem,

$\rm\implies \:DF \: \parallel \: BC$

$\rm\implies \:DF \: \parallel \:CE - - - (1)$

Also, In triangle ABC,

• D is the midpoint of AB.

• E is the midpoint of BC.

So, By midpoint theorem,

$\rm\implies \:DE \: \parallel \:AC$

$\rm\implies \:DE \: \parallel \:CF - - - (2)$

From equation (1) and (2), we have

$\rm\implies \:DE \: \parallel \:CF \: \: and \: \:DF \: \parallel \: CE$

$\rm\implies \:DECF \: is \: a \: parallelogram$

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Midpoint Theorem :- This theorem states that the line segment joining the midpoints of two sides of a triangle is parallel to third side and equals to half of it.

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Additional Information :-

1. If one pair of opposite sides of a quadrilateral are equal and parallel, then quadrilateral is a parallelogram.

2. Diagonals of rhombus are unequal and bisects each other at right angles.

3. Diagonals of a square are equal and bisects each other at right angles.

Answer 2

PROOF :-

Given, In ∆ABC

D is the midpoint of AB

E is the midpoint of BC

F is the midpoint of AC

Now, In ∆ABC,

D is the midpoint of AB

F is the midpoint of AC.

So, By midpoint theorem,

$\mapsto \: \tt DF \parallel BC$

$\mapsto \: \tt DF \parallel CE \: \: \: \underline{ \: \: \: \: \: \: \: \: \: \: } \: \boxed{1}$

Also, In ∆ABC,

D is the midpoint of AB.

E is the midpoint of BC.

So, By midpoint theorem,

$\mapsto \: \tt DE \: || \: AC$

$\mapsto \: \tt DE \: || \: CF \: \: \underline{ \: \: \: \: \: \: \: \: \: \: \: } \: \boxed{2}$

From equation (1) and (2), we have

$\tt \longmapsto \: \: DE \: \: || \: \: CF \: \: and \: \: DF \: \: || \: \: CE$

$\bf \therefore \: DECF \: \: is \: \: \: a \: \: parallelogram$