`P` and `Q` are points on the sides `C A` and `C B` respectively of ` A B C`, right-angled at `C`. Prove that `A Q^2+B P^2=A B^2+P Q^2`.

In Figure, Pa n dQ are the midpoints of the sides C Aa n dC B respectively of A B C right angled at C. Prove that 4(A Q^2+B P^2)=5A B^2 .

In a right triangle A B C right-angled at C ,\ P and Q are the points on the sides C A and C B respectively, which divide these sides in the ratio 2:1 . Prove that 9AQ^2=9AC^2+4BC^2

In a right triangle A B C right-angled at B , if P a n d Q are points on the sides A Ba n dA C respectively, then (a) A Q^2+C P^2=2(A C^2+P Q^2) (b) 2(A Q^2+C P^2)=A C^2+P Q^2 (c) A Q^2+C P^2=A C^2+P Q^2 (d) A Q+C P=1/2(A C+P Q)dot

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) .

P and Q are points on sides A B and A C respectively of A B C . If A P=3c m , P B=6c m , A Q=5c m and Q C=10 c m , show that B C=3\ P Q .

In a A B C ,\ E\ a n d\ F are the mid-points of A C\ a n d\ A B respectively. The altitude A P to B C intersects F E\ at Q . Prove that A Q=Q P .

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 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 .

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=4c m ,\ \ P B=6c m and P Q=3c m , determine B C .

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