Question

12. In quadrilateral ABCD, AC = BD and AC I BD.
If P, Q, R and S are the midpoints of the
sides AB, BC, CD and DA respectively, prove
that PQRS is a square.​

Answer 1

Step-by-step explanation:

Solution :

Given In a quadrilateral ABCD, P, Q, R and S are the mid-points of sides AB, BC, CD and DA, respectively. Also, AC = BD

To prove PQRS is a rhombus.

Proof In ΔADC, S and R are the mid-points of AD and DC respectively. Then, by mid-point theorem.

" "SR||AC and SR=(1)/(2)AC" "...(i)

In DeltaABC, P and Q are t he mid-points of AB and BC respectively. Then, by mid-point theorem

" "PQ||AC and PQ=(1)/(2)AC" "...(ii)

From Eqs. (i) and (ii), " "SR=PQ=(1)/(2)AC" "...(iii)

Similarly, in DeltaBCD, " "RQ||BD and RQ=(1)/(2)BD" "...(iv)

And in DeltaBAD, " "SP||BD and SP=(1)/(2)BD" "...(v)

From Eqs. (iv) and (v), " "SP=RQ=(1)/(2)BD=(1)/(2) AC" " [given, AC=BD]...(vi)

From Eqs. (iii) and (vi), " "SR=PQ=SP=RQ

It shows that all sides of a quadrilateral PQRS are equal.

Hence, PQRS is a rhombus. " " Hence proved.

Answer 2

Answer:

sorry dear it is hard to do....