Let `A(p^2,-p),B(q^2, q),C(r^2,-r)` be the vertices of triangle ABC. A parallelogram AFDE is drawn with D,E, and F on the line segments BC, CA and AB, respectively. Using calculus, show that the maximum area of such a parallelogram is `1/2(p+q)(q+r)(p-r)`.

P,Q,R are the points on the sides AB, BC and CA respectively of triangle ABC such that AP:PB=BQ:QC=AR:RC=1:2. Show that PBQR is a parallelogram.

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.

P and Q are any two points lying on the sides DC and AD respectively of a parallelogram ABCD . Show that a r\ (A P B)\ =\ a r\ (B Q C) .

Let P be an interior point of a triangle ABC , Let Q and R be the reflections of P in AB and AC , respectively if Q .A , R are collinear then angle A equals -

p\ A N D\ q are any two points lying on the sides D C\ a n d\ A D respectively of a parallelogram A B C Ddot\ Show that a r( A P B)=a r\ ( B Q C)dot

In Delta P Q R , if P Q=Q R and L ,M and N are the mid-points of the sides P Q ,Q R and R P respectively. Prove that L N=M Ndot

If P(2,\ 4),\ \ Q(0,\ 3),\ \ R(3,\ 6) and S(5,\ y) are the vertices of a parallelogram P Q R S , then the value of y is (a) 7 (b) 5 (c) -7 (d) -8

A B C D is a parallelogram. P is the mid-point of A BdotB D\ a n d\ C P intersect at Q such that C Q : Q P=3: 1. If a r\ ( P B Q)=10\ c m^2, find the area of parallelogram A B C D

In Fig. 9.32, ABCD is a parallelogram and BC is produced to a point Q such that A D\ =\ C Q . If AQ intersect DC at P, show that a r\ (B P C)\ =\ a r\ (D P Q)dot

The side AB of a parallelogram ABCD is produced to any point P. A line through A and parallel to CP meets CB produced at Q and then parallelogram PBQR is completed. Show that a r\ (A B C D)\ =\ a r\ (P B Q R) .