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.

tq ​

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

Construction :- Join DE and DF.

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 concluded that

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

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

$\rm\implies \:DECF \: is \: a \: parallogram.$

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Basic Concept Used :-

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.

4. Diagonal of parallelogram divides it in to two triangle of equal areas.