Question

In the adjoining figure, ABCD is a parallelogram. It circumscribes the circle with centre T. Points E, F, G, H are touching points. If AE = 4.5, EB = 5.5, find AD

Answer 1

Let the values of DH and CF be x and y respectively. 

[ AE = AH = 4.5 

BE = BF = 5.5 

DH = DG = x 

CF = CG = y ] [Tangent segment theorem] □ABCD is a parallelogram. [Given]

∴ AB = CD [Opposite sides of a parallelogram] 

∴ AE + BE = DG + CG [A – E – B, D – G – C] 

∴ 4.5 + 5.5 = x + y ∴ x + y = 10 ……….. (i) 

Also, AD = BC [Opposite sides of a parallelogram] 

∴ AH + DH = BF + CF [A – H – D, B – F – C] 

∴ 4.5 + x = 5.5 + y 

∴ x – y = 1 ………… (ii) Adding equations (i) and (ii), we get 

2x = 11 

∴ x = 11/2 = 5.5 

∴ AD = AH + DH [A – H – D] 

= 4.5 + 5.5 

∴ AD = 10 units