Geometry:tangent to circle

# Tangent to Circle

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A Tangent to Circle is a line that intersects the circle in exactly one point.

The point is called the point of contact of the tangent and the line is said to touch the circle at this point.

The word Tangent to Circle is originated from the Latin word
TANGREE which means "to touch".

The point of contact is the only point which is common to the tangent and the circle, so this point of contact is nearest to the center of the circle.

Some Properties of a tangent to the circle
1) A tangent to the circle is perpendicular to the radius through the point of contact. If AB is a tangent to the circle at P and OP is a radius then OP ⊥ AB
The converse of the above result is also true.
If radius OP ⊥ AB then AB is a tangent to the circle at P.
Length of a Tangent :

2) The length of two of tangents drawn from an external point to a circle are equal. If AP and AQ are two tangents to the circle then AP = AQ
3) If two tangents are drawn to a circle from an external point,
then 1) they subtend equal angles at the center. 2) they are equally inclined to the segment, joining the center to that point. If AP and AQ are tangents to the circle then ∠AOP = ∠AOQ and
∠OAP = ∠OAQ
Some solved examples on above result

1) A point P is 13 cm from the center of the circle. The length of the tangent drawn from P to the circle is 12 cm. Find the radius of the circle.

Solution :
Since the tangent to a circle is perpendicular to the radius through the point of contact. ∴ ∠OTP = 90
0
By Pythagorean theorem,

c
2 = a 2 + b 2

13
2 = OT 2 + 12 2

169 = OT
2 + 144

OT
2 = 169 – 144 = 25

⇒ OT = 5 cm

Hence, the radius of the circle is 5 cm.

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2) In the given figure, if AB = AC prove that BE = CE.

Given : AB = AC

Prove that : BE = EC Statements Reasons 1) AD = AF 1) The length of two of tangents drawn from an external point to a circle are equal. 2) BD = BE 2) The length of two of tangents drawn from an external point to a circle are equal. 3) CE = CF 3) The length of two of tangents drawn from an external point to a circle are equal. 4) AB = AC 4) Given 5) AB - AD = AC - AD 5) Subtract AD on both sides. 6) AB - AD = AC - AF 6) From (1) 7) BD = CF 7) By subtraction property 7) BE = CF 7) From (2) 8) BE = CE 8) From (3)

Circles

Circles
Parts of Circle
Arc and Chords
Equal Chords of a Circle
Arc and Angles
Cyclic Quadrilateral
Tangent to Circle

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