`A B C D` is a trapezium in which `A B D C` . `P` and `Q` are points on sides `A D` and `B C` such that `P Q A B` . If `P D=18` , `B Q=35` and `Q C=15` , find `A D` .

In A B C , P and Q are points on sides A B and A C respectively such that P Q ||B C . If A P=3c m , P B=5c m and A C=8c m , find A Q .

In a A B C , P and Q are point on the sides A B and A C respectively, such that P Q// B C . If A P=2. 4 c m , A Q=2c m , Q C=3c m and B C=6c m , find A B and P Q .

P Q R S is a trapezium having P S\ a n d\ Q R as parallel sides. A is any point on P Q\ a n d\ B is a point on S R such that A B || Q R . If area of /_\P B Q is 17\ c m^2, find the area of \ /_\A S R

In a triangle A B C , let P and Q be points on AB and AC respectively such that P Q || B C . Prove that the median A D bisects P Q .

In Figure, A B C D is a parallelogram in which P is the mid-point of D C\ a n d\ Q is a point on A C such that C Q=1/4A C . If P Q produced meets B C\ a t\ R , prove that R is a mid-point of B C .

In fig.22, A B C is right angled at Adot\ Q\ \ a n d\ R are points on line B C and P is a point such that Q P||A C and R P||A B . Find /_Pdot

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.

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 B C ,D is the mid-point of A B ,P is any point of B CdotC Q P D meets A B in Q . Show that a r( B P Q)=1/2a r( A B C)dot

A B C is a triangle in which A B=A C\ a n d\ P is a point on A C . Through C a line is drawn to intersect B P produced at Q such that /_A B Q=\ /_A C Q . Prove that: /_A Q C=90^@+1/2/_B A C