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Question
In the given figure, ▢ABCD is a parallelogram. It circumscribes the circle with centre T. Point E, F, G, H are touching points. If AE = 4.5, EB = 5.5, find AD.
Solution
ABCD is a parallelogram.
∴ AB = CD .....(1) (Opposite sides of parallelogram are equal)
AD = BC .....(2) (Opposite sides of parallelogram are equal)
Tangent segments drawn from an external point to a circle are congruent.
AE = AH .....(3)
DG = DH .....(4)
BE = BF .....(5)
CG = CF .....(6)
Adding (3), (4), (5) and (6), we get
AE + BE + CG + DG = AH + DH + BF + CF
⇒ AB + CD = AD + BC .....(7)
From (1), (2) and (7), we have
2AB = 2AD
⇒ AB = AD
∴ AD = AB = AE + EB = 4.5 + 5.5 = 10 units
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