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` .

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 .

BL and CM are medians of a triangle ABC right angled at A. Prove that 4(B L^2+C M^2)=5B C^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 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 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 .

A B C is an isosceles right triangle right-angled at C . Prove that A B^2=2A C^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

In a triangle, P ,\ Q\ a n d\ R are the mid-points of sides B C ,\ C A\ a n d\ A B respectively. If A C=21\ c m ,\ B C=29 c m\ a n d\ A B=30 c m , find the perimeter of the quadrilateral A R P Q

A B C D is a quadrilateral; P , Q , Ra n dS are the points of trisection of side A B ,B C ,C Da n dD A respectively and are adjacent to Aa n dC ; prove that P Q R S is parallelogram.

In Figure, M ,N and P are the mid-points of A B ,A C and B C respectively. If M N=3c m ,N P=3. 5c m and M P=2. 5c m , calculate B C ,A B and A Cdot