Question

"Question 1 ABCD is a quadrilateral in which P, Q, R and S are mid-points of the sides AB, BC, CD and DA (see the given figure). AC is a diagonal. Show that: (i) SR || AC and SR = AC (ii) PQ = SR (iii) PQRS is a parallelogram.

Class 9 - Math - Quadrilaterals Page 150"

Answer 1

1)(i)consider triangle ACD
Using mid point theorem,
S is midpt. of AD
R is midpt. of DC 
Midpt. theorem states that the midpts. of 2 sides when joined are parallel to the 3rd side and are equal to (3rd side/2)
therefore SR is parallel to AC
(ii) I suppose your question would be SR=(AC/2)
Using the above theorem, (3rd side/2)=length of line connecting mid pts.,
SR=AC/2
(2)PQ=AC/2
    SR=AC/2
    As Euclid's Axiom states that things which are halves of the same thing         are equal,
PQ=SR=AC/2 
    PQ is parallel to AC which is parallel to SR
PQ is parallel to SR
(3) Construct diagonal BD of the quad.ABCD. By using the mid pt. theorem,
     RQ=BD/2
     SP=BD/2
     RQ=SP=BD/2(Euclid's axiom) RQ is parallel to BD which is parallel to SP
      RQ is parallel to SP.
PQ=SR=AC/2 
    PQ is parallel to AC which is parallel to SR
PQ is parallel to SR(PROVED ABOVE)
therefore PQRS is a parallelogram as the opposite parallel sides are equal.
please mark as brainliest
      

Answer 2

A quadrilateral is a parallelogram if

i)Its opposite sides are equal

ii) its opposite angles are equal

iii) diagonals bisect each other

iv) a pair of opposite sides is equal and parallel.

 

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Solution:

Given:

P,Q, R & S are the midpoints of the sides AB, BC ,CD and DA

 ∴ AP=BP

BQ=CQ

CR=DR

AS=DS

 

To show:

 (i) SR || AC and SR =1/2 AC 

(ii) PQ = SR 

(iii) PQRS is a parallelogram.

 

Proof:

 (i)  In △ADC,

R is the mid point of DC and S is the mid point of AD.

 

We know that the line segment joining the midpoints of any two sides of a triangle is parallel to the third side and equal to half of it.

Thus by mid point theorem, 

SR∥AC...........(i)

 & SR= ½ AC........(ii)

 

(ii) Similarly, In △ABC,
P is the mid point of AB and Q is the mid point of BC.

Thus by mid point theorem, 

PQ∥AC ..........(iii)

& PQ= 1/2AC........(iv)

 

From eq ii & iv

SR=PQ=1/2AC...... (V)

(iii)

From eq i & iii

 PQ||SR & From eq. V

 PQ = SR

 Since, a pair of opposite sides of a quadrilateral PQRS is equal and parallel.

So, PQRS is a parallelogram.

 

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Hope this will help you.....