Question

A pair of tangents to a circle which is inclined to each other at an angle of 60° are drawn at ends of two radii. The angle between these radii must be: / एक वृत्त की दो स्पर्श रेखाएँ एक दूसरे के साथ 60° का कोण बनाती हैं। वृत्त की जिन दो त्रिज्याओं के अंत बिंदुओं पर स्पर्श रेखाएं खीचीं गयी है, वे आपस में एक दूसरे के साथ कितना कोण बनाती हैं? *

30°

60°

140°

120°​

Answer 1

Given :- A pair of tangents to a circle which is inclined to each other at an angle of 60° are drawn at ends of two radii. The angle between these radii must be: / एक वृत्त की दो स्पर्श रेखाएँ एक दूसरे के साथ 60° का कोण बनाती हैं। वृत्त की जिन दो त्रिज्याओं के अंत बिंदुओं पर स्पर्श रेखाएं खीचीं गयी है, वे आपस में एक दूसरे के साथ कितना कोण बनाती हैं ? 

A) 30°

B) 60°

C) 140°

D) 120°

Solution :-

from image , we have,

• OA and OC = radius of circle with centre O .
• AB and CB tangents from circle meets at B .
• ∠ABC = 60° (given) .

we know that,

• Tangent to a circle is always perpendicular to the radius.

So,

• ∠OAB = 90°
• ∠OCB = 90° .

Now, in quadrilateral OABC , we have,

→ ∠ABC + ∠OAB + ∠OCB + ∠AOC = 360° (angle sum property.)

→ 60° + 90° + 90° + ∠AOC = 360°

→ 240° + ∠AOC = 360°

→ ∠AOC = 360° - 240°

→ ∠AOC = 120° (Option D) (Ans.)

Hence, The angle between these radii must be 120°.

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