Question

tangents are drawn to a circle from an external point of a distance of 13 cm away from the centre of the tangents if the tangents are inclinedas anche angle 60° then what is the radius of the circle.

Answer 1

If the tangents are inclined as at angle 60° and the external point is at a distance of 13 cm away from the centre then the radius of the circle is 6.5 m.

Step-by-step explanation:

Step 1:  

Referring to the figure attached below, let’s consider ∆ ABO and ∆ ACO, we have

∠ABO = ∠ACO = 90° …….. [∵ A tangent to a point on the circle is perpendicular to its radius]

AO = AO ….. [common hypotenuse]

BO = CO …… [radius of the circle]

∴ By RHS congruence, ∆ ABO ≅ ∆ ACO

∴ ∠BAO = ∠CAO = ½ ∠BAC = 30° …… [by C.P.C.T and  also ∠BAC is given as 60°]

Step 2:

Now, consider ∆ ABO, applying trigonometry properties of triangles, we get

sin θ = $\frac{Perpendicula}{Hypotenuse} = \frac{OB}{AO}$

⇒ sin 30° = $\frac{OB}{13}$ ….. [since the distance from the external points from, which the tangents are drawn, to the centre of the circle, AO is given as 13 m]

⇒ ½ =  $\frac{OB}{13}$

⇒ OB =  $\frac{13}{2}$

⇒ OB = 6.5 m

Thus, the radius of the circle is 6.5 m.

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