In the right triangle B is a point on AC such that AB+AD=BC+CD if AB= x.BC= h and CD= d then find x

In the right triangle B is a point on AC such that AB+AD=BC+CD if AB=

1.	In the right triangle B is a point on AC such that AB+AD=BC+CD if AB= x.BC= h and CD= d then find x
Answer» Here we are given that{tex}\\therefore \\quad A B + A D = B C + C D{/tex}or,\xa0{tex}AD = BC+CD-AB{/tex}or,\xa0{tex}AD =h+d-x{/tex}In the right angled\xa0{tex}\\Delta A C D,{/tex}\xa0{tex}A D ^ { 2 } = A C ^ { 2 } + D C ^ { 2 }{/tex}or,\xa0{tex}( h + d - x ) ^ { 2 } = ( x + h ) ^ { 2 } + d ^ { 2 }{/tex}or,\xa0{tex}( h + d - x ) ^ { 2 } - ( x + h ) ^ { 2 } = d ^ { 2 }{/tex}\xa0{tex}( h + d - x - x - h ) ( h + d - x + x + h ) = d ^ { 2 }{/tex}\xa0{tex} Because \\ a^2-b^2=(a-b)(a+b){/tex}or,\xa0{tex}( d - 2 x ) ( 2 h + d ) = d ^ { 2 }{/tex}or,\xa0{tex}2 h d + d ^ { 2 } - 4 h x - 2 x d = d ^ { 2 }{/tex}or,\xa0{tex}2hd = 4hx+2xd{/tex}\xa0{tex}2hd= 2x(2h+d){/tex}Hence {tex}x = \\frac { h d } { 2 h + d }{/tex}