In the figure, `square ABCD` is a parallalogram. It circumscribes the cirlcle with centre T. Point E,F,G,H are touching points. If AE = 4.5 ,EB = 5.5 , find AD.

Answer

AD = 10

Explanation

AE = 4.5 and EB = 5.5 …(Given )
AH = AE = 4.5
BF = BE = 5.5
Let DH = DG = x
CG = CF = y ….(Tangent segment theorem )
`square ` theorem is a parallelogram ….(Given )
`:. AB = CD ` ….( Opposite sides of a parallelogram are equal )
`:.` AE + EB = CG + GD ....(A-E-B and C-G-D )
`:. 4.5 + 5.5 = y + x `
`:. x + y = 10` ...(1)
AD = BC ...( Opposite sides of a parallelogram are equal )
`:. AH + HD= BF + FC ` ...( A-H - D and B-F-C )
`:. 4.5 + x = 5.5 + y `
`:. x - y = 5.5 - 4.5 `
`:. x-y = 1 ` ......(2)
Adding (1) and (2), we get, ,brgt `x+ y = 10`
`x-y = 1 `
`bar ( 2x = 11 )`
`:. x = ( 11)/(2)`
`:. x = 5.5 `
`:. DG = DH = 5.5 `
`:. AD = AH + HD ` ( A-H -D)
`:. AD = 4.5 + 5.5 `
`:. AD = 10`