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Question

ABCD is a rectangle formed by the points A(-1,-1) , B(-1,4) , C(5,4) andD(5,-1) . P,Q,R and S are the midpoints of AB,BC,CD and DA respectively. Is the quadrilateral PQRS a square? a rectangle? or a rhombus? Justify your answer.


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Solution

Step 1:Finding the midpoints P,Q,R and S:

P,Q,R and S are the midpoints of AB,BC,CD and DA respectively.

Coordinate of P=-1+(-1)2,-1+42=-22,32=-1,32

Coordinate of Q=-1+52,4+42=42,82=2,4

Coordinate of R=5+52,4+(-1)2=102,32=5,32

Coordinate of S=5+(-1)2,-1+(-1)2=42,-22=2,-1

Step 2: Finding the length of PQ,QR,RS and SP

Using distance formula =x2-x12+y2-y12

Length of PQ

=x2-x12+y2-y12=2-(-1)2+4-322=2+12+8-322=32+522=9+254=36+254=614=612

Length of QR

=x2-x12+y2-y12=2-52+4-322=-32+8-322=9+522=9+254=36+254=614=612

Length of RS

=x2-x12+y2-y12=2-52+-1-322=-32+-2-322=9+-522=9+254=36+254=614=612

Length of SP

=x2-x12+y2-y12=2-(-1)2+-1-322=2+12+-2-322=(3)2+-522=9+254=36+254=614=612

Step 3: Finding the length of diagonals PR and SQ:

Length of PR

=x2-x12+y2-y12=5-(-1)2+32-322=5+12+3-322=62+022=36+0=36=6

Length of SQ

=x2-x12+y2-y12=2-22+4-(-1)2=02+4+12=0+52=25=5

Step 4: Finding PQRS is a square or rectangle or rhombus:

As , the lengths PQ=QR=RS=SP=612,

But, the lengths of diagonals are not equal as,

PR=6,SQ=5PRSQ

Hence, PQRS is a rhombus.


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