Length of a tangent

The length of the tangent from the point (Say P) to the circle is defined as the segment of the tangent from the external point P to the point of tangency I with the circle. In this case, PI is the tangent length.

Lengths of tangents drawn from an external point

Theorem: Two tangents are of equal length when the tangent is drawn from an external point to a circle.

Tangents to a circle from an external point

PT1 = PT2

Example

Find the length of AB in the given circle, which is the chord in the outer circle and tangent to the inner circle. The radius of the inner and outer circle is 6 cm and 10 cm respectively.

Solution

Given

Radius of the inner circle (r) = 6 cm

Radius of outer circle (R) = 10 cm

As the Point T which is the tangent point is the midpoint of the chord, AT = TB

As radius is perpendicular to the tangent,

So is a right angle triangle and we can use Pythagoras theorem.

OB2 = OT2 + TB2

TB= OB2 – OT2

= 102 – 62

= 100 – 36

 TB2 = 64

TB = 8 cm

AB = TB + AT

AB = 8 + 8 (AT = BT)

AB = 16 cm

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