Question

10. ABCD is a square. P, Q, M, and N are the midpoints of the sides on which they lie. Prove that PQ = MN​

Answer 1

We know that ABCD is a square and P,Q are mid-points of DA and BC

⇒ AD=BC [ Sides of square are equal ]

⇒ DP=PA [ P is the mid point ]

⇒ CQ=QB [ Q is the mid point ]

⇒ DA=2AP ---- ( 1 )

⇒ CB=2CQ ---- ( 2 )

From ( 1 ) and ( 2 ),

⇒ 2×AP=2×CQ

⇒ AP=CQ ---- ( 3 )

Now, in △PAB and △QCD

⇒ AP=CQ [ From ( 3 ) ]

⇒ ∠PAB=∠QCD [ Angles of a square is 90

o

. ]

⇒ AB=CD [ Sides of square ]

⇒ △PAB≅△QCD [ By SAS property ]

⇒ PB=QD [ By CPCT ]

Answer 2

Answer:

PQ=MN

Step-by-step explanation:

Since ABCD is a square all sides are equal.

1/2 of AB is BM and 1/2 of AD is PD. Hence BM=PD

anglePDQ=angleMBN [since all angles are 90 degrees in a square]

1/2 of BC is BN and 1/2 of DCis DQ. Hence BN=DQ

Therefore by SAS congruency triangle BMN is congruent to triangle DPQ

By CPST PQ=MN

Hence proved