Coordinated Trouble #4

Geometry Level 3

G i v e n a t r i a n g l e h a v i n g v e r t i c e s A ( 2 , 11 2 ) , B ( 1 , 1 ) , C ( 13 , 2 3 ) . I f X = A B 2 + A C 2 & Y = A D 2 + D C 2 . F i n d X Y , i f D i s t h e m i d p o i n t o f B C . Given\quad a\quad triangle\quad having\quad vertices\quad \\ A(2,\frac { 11 }{ 2 } ),\quad B(1,1),\quad C(13,\frac { 2 }{ 3 } ).\\ If\quad X={ AB }^{ 2 }+{ AC }^{ 2 }\quad \& \quad Y={ AD }^{ 2 }+{ DC }^{ 2 }.\\ Find\quad \frac { X }{ Y } ,if\quad D\quad is\quad the\quad midpoint\quad of\quad BC.


The answer is 2.

Soummo Mukherjee by Soummo Mukherjee , 25, India

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1 solution

Samyak Jain
Nov 25, 2014

D is the midpoint of BC therefore AD is the median BY USING APOLLONIUS THEOREM, AB^{2}+AC^{2}=2(AD^{2}+DC^{2}) THEREFORE X=2Y X/Y=2

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