A CHALLENGING QUESTION
CLASS IX STUDENTS
BRAINLIEST FOR SURE
Q. Let ABCD is a Parallelogram in which X and Y are the Mid-Points of AB and DC respectively. If AY and DX intersect in P while CX and BY intersect In Q, show that quadrilateral PXQY is a parallelogram.
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shreyansgarg2004:
I remember this. This is from this year's IMO class 9 level 1 but I won't be able to answer as I am not at home right now. I was just wondering here. Only thing I can say is to prove congruency of opp. Triangles and it will be easily done
Given: X and Y are the mid-points of opposite sides AB and DC of a parallelogram ABCD.
AX = XB and DY = YC
To Prove: PXQY is a parallelogram.
Proof: AB = DC
... AB =  DC
XB = DY......(i)
Also AB ïï DC...... (opp. sides of a ïïgm)
XB ïï DY.....(ii)
Since in quadrilateral XBYD, XB = DY and XB ïï DY
... XBYD is a parallelogram.
... DX ïï YB ⇒ PX ïï YQ......(iii)
Similarly we can prove, PY ïï XQ ......(iv)
From (iii) and (iv), we get PXQY is a parallelogram.
AX = XB and DY = YC
To Prove: PXQY is a parallelogram.
Proof: AB = DC
... AB =  DC
XB = DY......(i)
Also AB ïï DC...... (opp. sides of a ïïgm)
XB ïï DY.....(ii)
Since in quadrilateral XBYD, XB = DY and XB ïï DY
... XBYD is a parallelogram.
... DX ïï YB ⇒ PX ïï YQ......(iii)
Similarly we can prove, PY ïï XQ ......(iv)
From (iii) and (iv), we get PXQY is a parallelogram.
Answers
Answered by
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HEY MATE!!!
GIVEN:- ABCD IS A PARALLELOGRAM
X AND Y ARE THE MID POINT OD AB AND DC
TO PROOF :- PXQY IS A PARALLELOGRAM
SOLUTION :- IN TRIANGLE APD
A + D + P = 180 DEGREE ( ASP OF TRIANGLE )
1/2 A + 1/2 D + C = 180
1/2 ( A + D) = 180 adjacent side of parallelogram is 180
A + D = 90 DEGREE
90 + P = 180
P = 90 DEGREE
SIMILLARLY in other triangle we will show that angle Q is 90 degree
since opposite angle are equal and 90 degree its a rectangle and all rectangle is a parallelogram.
therefore PXQY IS A PARALLELOGRAM
if any doudt plzz let me know
diagram is attach
GIVEN:- ABCD IS A PARALLELOGRAM
X AND Y ARE THE MID POINT OD AB AND DC
TO PROOF :- PXQY IS A PARALLELOGRAM
SOLUTION :- IN TRIANGLE APD
A + D + P = 180 DEGREE ( ASP OF TRIANGLE )
1/2 A + 1/2 D + C = 180
1/2 ( A + D) = 180 adjacent side of parallelogram is 180
A + D = 90 DEGREE
90 + P = 180
P = 90 DEGREE
SIMILLARLY in other triangle we will show that angle Q is 90 degree
since opposite angle are equal and 90 degree its a rectangle and all rectangle is a parallelogram.
therefore PXQY IS A PARALLELOGRAM
if any doudt plzz let me know
diagram is attach
Attachments:
Answered by
1
Given: X and Y are the mid-points of opposite sides AB and DC of a parallelogram ABCD.
=>AX = XB and DY = YC
To Prove: PXQY is a parallelogram.
Proof: AB = DC
... AB =  DC
XB = DY......(i)
Also AB = DC...... (opp. sides of a \\gm)
XB= DY.....(ii)
Since in quadrilateral XBYD, XB = DY and XB = DY
... XBYD is a parallelogram.
... DX = YB ⇒ PX =YQ......(iii)
Similarly we can prove, PY = XQ ......(iv)
From (iii) and (iv), we get PXQY is a parallelogram.
Attachments:
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