In parallelogram A B C D, A C=10 and B D=28 . The points K and L in the plane of A B C D move in such a way that A K=B D and B L=A C . Let M and N be the midpoints of C K and D L , respectively. What is the maximum value of ^2( BMD / 2)+ ^2( ANC / 2) ?

Step-by-step solution

CK ₁=18 aligned & OM ₁=5+9=14 & BM ₁ D =90^ ( in semicircle ) & Similarly, CK ₂=10+28=38 & CM ₂=19 & AM ₂=9 & OM ₂=5+9=14 & BM ₂ D =90^ ( in semicircle ) aligned If K ₁ or K ₂ moves BMD will increase By observation : BMD 2 ( BMD 2 ) For maximum BMD =90^ ^2 ( BMD 2 )=1 aligned & BL ₁=10, OL ₁=4, DL ₁=18, DN ₁=9, ON ₁=5 & AN ₁ C = 2 ( in semicircle ) aligned For L2 point, aligned & DL ₂=14+14+10 & DL ₂=38 & DN ₂=19 & ON ₂=5 aligned AN ₂ C = 2 ( in semicircle ) Now if L₁ or L₂ moves, ANC ANC 2 ANC 2 (Observation) Henc