In the Given Figure, Two Tangents Ab and Ac Are Drawn to a Circle with Centre O Such that ∠Bac = 120°. Prove that Oa = 2ab. - Mathematics

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Short Note

In the given figure, two tangents AB and AC are drawn to a circle with centre O such that ∠BAC = 120°. Prove that OA = 2AB.

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Solution

Consider Δ OAB and Δ OAC.

We have,

OB = OC (Since they are radii of the same circle)

AB = AC (Since length of two tangents drawn from an external point will be equal)

OA is the common side.

Therefore by SSS congruency, we can say that Δ OAB and Δ OAC are congruent triangles.

Therefore,

∠OAC = ∠OAC

It is given that,

`∠OAB +∠OAC=120^o`

`2∠OAB=120^o`

`∠OAB=60^o`

We know that,

`cos∠OAB =(AB)/(OA) `

`cos 60^o =(AB)/(OA) `

We know that,

`cos 60^o =1/2`

Therefore,

`1/2=(AB)/(OA)`

OA = 2AB

  Is there an error in this question or solution?
Chapter 8: Circles - Exercise 8.2 [Page 38]

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RD Sharma Class 10 Maths
Chapter 8 Circles
Exercise 8.2 | Q 33 | Page 38

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