Question

D. E and F are the mid points of the sides
AB, BC and CA respectively of triangle ABC, AE
meets DF at O. P and Q are the mid-points
of OB and OC respectively. Prove that DPQF
is a parallelogram.​

Answer 1

Given:

D, E and F are the midpoints of the sides  AB, BC and CA respectively of triangle ABC

AE  meets DF at O

P and Q are the mid-points  of OB and OC respectively

To Prove:

DPQF is a parallelogram

Solution:

$\boxed{\bold{\underline{MIDPOINT\:THEOREM}}}:$ This theorem states that the line-segment joining the midpoints of any two sides of a triangle is parallel to its third side and is half the length of the third side.

In Δ ABC , using the above midpoint theorem, we have

D & F are the midpoints of side AB and AC respectively

∴ DF // BC and DF =  $\frac{1}{2} BC$ ...... (i)

In Δ OBC , using the above midpoint theorem, we have

P & Q are the midpoints of side OB and OC respectively

∴ PQ // BC and PQ =  $\frac{1}{2} BC$ ...... (ii)

From (i) and (ii), we get

DF // BC // PQ

⇒ DF // PQ .... (iii)

and

DF =  $\frac{1}{2} BC$ = PQ

⇒ DF = PQ ..... (iv)

Similarly, using the midpoint theorems, we get

In Δ AOB, we have

DP // AO and DP =  $\frac{1}{2} AO$ ..... (v)

In Δ AOC , we have

FQ // AO and FQ =  $\frac{1}{2} AO$ ..... (vi)

From (v) and (vi), we get

DP // AO // FQ

⇒ DP // FQ .... (vii)

and

DP =  $\frac{1}{2} AO$ = FQ

⇒ DP = FQ ..... (viii)

Now, from (iii), (iv), (vii) & (viii), we get

DF // PQ & DP // FQ and DF = PQ & DP = FQ

⇒ Any quadrilateral having opposite sides parallel to each other and equal in length is known as a parallelogram.

∴ $\boxed{\bold{\underline{DPQF \:is\: a\: parallelogram }}}$

Hence proved

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Also View:

In triangle ABC, E and F are the midpoint of AB and AC respectively. If EF = 5.  then find BC.​

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State and prove mid point theorem with diagram.

https://brainly.in/question/7298842

Answer 2

Step-by-step explanation:

Given D. E and F are the mid points of the sides AB, BC and CA respectively of triangle ABC, AE  meets DF at O. P and Q are the mid-points  of OB and OC respectively. Prove that DPQF  is a parallelogram.

• So there is a triangle ABC,
• We get FD parallel to BC ------------1
• And BC = 2 FD (since D and F are midpoints by mid point theorem)
• So from triangle COB, we get
• PQ parallel to BC -----------------2
• Also BC = 2 PQ (since P and Q are midpoints by mid point theorem)
• So from triangle OAB
• PD is parallel to OA ---------------3
• Also OA = 2 PD (since P and D are midpoints by mid point theorem)
• So from triangle OCA,
• QF is parallel to OA----------------------4
• Also OA = 2 QF (since Q and F are midpoints by mid point theorem)
• From 1 and 2 we get
• FD = PQ and so FD is parallel to PQ
• From 3 and 4 we get
• PD = QF and PD is parallel to QF
• Therefore DPQF is a parallelogram

Reference link will be

https://brainly.in/question/1941453