In an isosceles ` A B C` , the base `A B` is produced both the ways to `P` and `Q` such that `A PxxB Q=A C^2dot` Prove that ` A P C B C Q` .

A B C D is a parallelogram. P is a point on A D such that A P=1/3\ A D\ a n d\ Q is a point on B C such that: C Q=1/3B Cdot Prove that A Q C P is a parallelogram.

In A B C and P Q R Figure, A B=P Q ,B C=Q R and C B and R Q are extended to X and Y respectively and /_A B X =/_P Q Ydot Prove that A B C~=P Q Rdot Figure

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

In A B C , P divides the side A B such that A P : P B=1:2 . Q is a point in A C such that P Q B C . Find the ratio of the areas of A P Q and trapezium B P Q C .

In A B C , /_A is obtuse, P B_|_A C and Q C_|_A B . Prove that: A BxxA Q=A CxxA P (ii) B C^2=(A CxxC P+A BxxB Q)

A B C D is a quadrilateral. A line through D , parallel to A C , meets B C produced in P as shown in Figure. Prove that a r\ ( A B P)=a r\ (Q u a ddotA B C D)dot

Triangles on the same base and between the same parallels are equal in area. GIVEN : Two triangles A B C and P B C on the same base B C and between the same parallel lines B C and A Pdot TO PROVE : a r( A B C)=a r( P B C) CONSTRUCTION : Through B , draw B D C A intersecting P A produced in D and through C , draw C Q B P , intersecting line A P in Q.

In Figure, A B C D\ a n d\ P Q R C are rectangles and Q is the mid-point of A C . Prove that D P=P C (ii) P R=1/2\ A C

In Figure, A B C D is a parallelogram in which P is the mid-point of D C and Q is a point on A C such that C Q=1/4A Cdot If P Q produced meets B C at Rdot Prove that R is a mid-point of B Cdot

In A A B C ,\ P\ a n d\ Q are respectively the mid-points of A B\ a n d\ B C and R is the mid-point of A Pdot Prove that: a r\ ( P B Q)=\ a r\ (\ A R C)