Question

PQRS is a square. A,B, C&D are mid points of sides on which they lie. prove that AB=DC ​

Answer 1

Here, we are joining A and C.

In ΔABC

P is the mid point of AB

Q is the mid point of BC

PQ∣∣AC [Line segments joining the mid points of two sides of a triangle is parallel to AC(third side) and also is half of it]

PQ=

2

1

AC

In ΔADC

R is mid point of CD

S is mid point of AD

RS∣∣AC [Line segments joining the mid points of two sides of a triangle is parallel to third side and also is half of it]

RS=

2

1

AC

So, PQ∣∣RS and PQ=RS [one pair of opposite side is parallel and equal]

In ΔAPS & ΔBPQ

AP=BP [P is the mid point of AB)

∠PAS=∠PBQ(All the angles of rectangle are 90

o

)

AS=BQ

∴ΔAPS≅ΔBPQ(SAS congruency)

∴PS=PQ

BS=PQ & PQ=RS (opposite sides of parallelogram is equal)

∴ PQ=RS=PS=RQ[All sides are equal]

∴ PQRS is a parallelogram with all sides equal

∴ So PQRS is a rhombus.

Answer 2

Step-by-step explanation:

Construction:- join SQ

PROOF:-

CD=1/2SQ(MIDPOINT THEOREM)--------(1)

AB=1/2SQ (MIDPOINT THEROEM)------(2)

FROM EQUATION (1) AND (2) WE GOT,

AB=1/2SQ=CD=1/2SQ

AB=CD (CANCELLING 1/2 SQ FROM BOTH SIDES)

HENCE PROVED