In Figure, `P Q R S` is a parallelogram. If `X\ a n d\ Y` are mid-points of `P Q\ a n d\ S R` respectively and diagonal `S Q` is joined. The ratio `a rX Q R Y:\ a r\ ( Q S R)=`
(a) 1:4
(b) 2:1
(c) 1:2
(d) 1:1

P Q R S is a parallelogram. P X\ a n d\ Q Y are respectively, the perpendiculars from P\ a n d\ Q to S R\ a n d\ R S produced. Prove that P X=Q Y

In Figure, X\ a n d\ Y are the mid-points of A C\ a n d\ A B respectively, Q P || B C\ a n d\ C Y Q\ a n d\ B X P are straight lines. Prove that a r\ (\ A B P)=\ a r\ (\ A C Q) .

In Figure, P Q R S is a square and T\ a n d\ U are, respectively, the mid-points of P S\ a n d\ Q R . Find the area of /_\ O T S\ if P Q=8\ c m .

In Figure, P Q > P R. Q S\ a n d\ R S are the bisectors of /_ Q and /_R respectively. Prove that S Q > S R .

In triangle P Q R , if P Q=R Q\ a n d\ L ,\ M\ a n d\ N are the mid-points of the sides P Q ,\ Q R\ a n d\ R P respectively. Prove that L N=M N .

Prove that the figure formed by joining the mid-points of the pair of consecutive sides of a quadrilateral is a parallelogram. OR ABC D is a parallelogram in which P ,Q ,R and S are mid-points of the sides A B ,B C ,C D and D A respectively. A C is a diagonal. Show that : P Q|| A C and P Q=1/2A C S R || A C and S R=1/2A C P Q=S R (iv) P Q R S is a parallelogram.

A B C D is a rhombus and P ,\ Q ,\ R ,\ S are the mid-points of A B ,\ B C ,\ C D ,\ D A respectively. Prove that P Q R S is a rectangle.

P Q R S is a parallelogram. P X and QY are respectively, the perpendiculars from P and Q to S R and R S produced. Prove that P X=Q Ydot

ABCD is a rhombus and P ,Q ,R ,S are the mid-points of A B ,B C ,C D ,D A respectively. Prove that P Q R S is a rectangle.

In Figure, T Q\ a n d\ T R are the bisectors of /_Q\ a n d\ /_R respectively. If /_Q P R=80^0a n d\ /_P R T=30^0, determine /_T Q R\ a n d\ /_Q T R