##### In figure, O is the center of a circle of radius 5 cm, T is a point such that OT = 13 and OT intersects the circle at E, if AB is the tangent to the circle at E, find the length of AB.

Given : A circle with center O and radius = 5cm T is a point, OT = 13 cm. OT intersects the circle at E and AB is the tangent to the circle at E .

To Find : Length of AB

OP PT [Tangent at a point on the circle is perpendicular to the radius through point of contact ]

By phythagoras theorem in OPT right angled at P

(OT)2 = (OP)2+ (PT)2

(13)2 = (5)2 + (PT)2

(PT)2= 169 - 25 = 144

PT = 12 cm

PT = TQ = 12 cm [Tangents drawn from an external point to a circle are equal]

Now,

OT = OE + ET

ET = OT - OE = 13 - 5 = 8 cm

Now, as Tangents drawn from an external point to a circle are equal .

AE = PA [1]

EB = BQ [2]

Also OE AB [Tangent at a point on the circle is perpendicular to the radius through point of contact ]

AEO = 90°

AEO + AET = 180° [By linear Pair]

AET = 90°

In AET By Pythagoras Theorem

(AT)2 = (AE)2 + (ET)2

[Here AE = PA as tangents drawn from an external point to a circle are equal]

(PT - PA)2 = (PA)2 + (ET)2

(12 - PA)2 = (PA)2 + (8)2 [from 1]

144 + (PA)2 - 24PA = (PA)2 + 64

24PA = 80

[3]

AET + BET = 180 [Linear Pair]

90 + BET = 180

BET = 90

In BET, By Pythagoras Theorem

(BT)2 = (BE)2+ (ET)2

(TQ - BQ)2 = (BQ)2 + (ET)2 [from 2]

(12 - BQ)2 = (BQ)2 + (8)2

144 + (BQ)2 - 24BQ = (BQ)2 + 64

24BQ = 80

[4]

So,

AB = AE + BE

AB = PA + BQ [From 1 and 2]

[From 3 and 4]

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