Question

$\red{1}$
Angle between tangents to a circle from an external point is 60° . Find the length of tangent if radius of circle is 5 cm . ​

Answer 1

$\large{\underline{\underline{\mathfrak{\sf{\blue{Answer-}}}}}}$

Length of tangent is 5√3 cm.

$\large{\underline{\underline{\mathfrak{\sf{\blue{Explanation-}}}}}}$

$\underline\bold\pink{\sf{Given-}}$

• Angle between tangents to a circle from an external point is 60°.
• Radius of circle i.e OP = OQ = 5cm

$\underline\bold\pink{\sf{To\:Find-}}$

• Length of tangent.

$\underline\bold\pink{\sf{Construction-}}$

• Join OA such that ∠PAO = ∠OAQ = 30°

$\underline\bold\pink{\sf{Solution-}}$

We know that, tangents drawn from external point are equal.

$\therefore$ $\sf\red{AP=AQ}$ _________(1)

We also know that, radius is always perpendicular to the tangent.

$\therefore$ ∠APO = AQO = 90°

Now, in right angled ∆APO,

TanØ = $\dfrac{P}{B}$

$\implies$ Tan30° = $\dfrac{OP}{AP}$

$\implies$ $\dfrac{1}{\sqrt3}$ = $\dfrac{5}{AP}$ ( Tan30° = $\dfrac{1}{\sqrt3}$ )

$\implies$ AP = 5√3

Also, AP = AQ ( from 1 )

Hence, AP = AQ = 5√3 cm.

_________________________

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Answer 2

Answer:

Step-by-step explanation:

\large{\underline{\underline{\mathfrak{\sf{\blue{Answer-}}}}}}

Length of tangent is 5√3 cm.

\large{\underline{\underline{\mathfrak{\sf{\blue{Explanation-}}}}}}

\underline\bold\pink{\sf{Given-}}

   Angle between tangents to a circle from an external point is 60°.

   Radius of circle i.e OP = OQ = 5cm

\underline\bold\pink{\sf{To\:Find-}}

   Length of tangent.

\underline\bold\pink{\sf{Construction-}}

   Join OA such that ∠PAO = ∠OAQ = 30°

\underline\bold\pink{\sf{Solution-}}

We know that, tangents drawn from external point are equal.

\therefore \sf\red{AP=AQ} _________(1)

We also know that, radius is always perpendicular to the tangent.

\therefore ∠APO = AQO = 90°

Now, in right angled ∆APO,

TanØ = \dfrac{P}{B}

\implies Tan30° = \dfrac{OP}{AP}

\implies \dfrac{1}{\sqrt3} = \dfrac{5}{AP} ( Tan30° = \dfrac{1}{\sqrt3} )

\implies AP = 5√3

Also, AP = AQ ( from 1 )

Hence, AP = AQ = 5√3 cm.