Circle MANIA

Geometry Level 2

Triangle PAB is formed by three tangents to circle with centre O

& angle-APB = 40 degree as shown in the figure,

Then angle-AOB equals...

70 55 50 45

Archiet Dev by Archiet Dev , 20, India

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2 solutions

Amitej Premkumar
Apr 27, 2014

Draw a construction line from O to tangent AB, let this point be C. Now, triangles OCB and ORB are similar by S-A-S test of similarity and angles COB and ROB are the same, let angle COB be 'a'. Similarly, we can prove that angles COA and TOA are the same and let angle COA be 'b'. In quad PROT, we have P as 40, R and T as 90, O as (2a + 2b), the sum of these angles in a quad is 360. On solving, we get (a+b) = 70, the required ans!

5 Helpful 1 Interesting 1 Brilliant 0 Confused
Vidit Kothari
Apr 27, 2014

I guessed the angle to be near to 90, coz the line AB is tangent to the circle.

But I couldn't figure out the exact reason/answer

1 Helpful 0 Interesting 0 Brilliant 0 Confused

we can construct the lines from centre to the tangents from outer point. now the angles substended by lines from centre and the point from intersecting tangents will be equal to the angle between the lines from intersecting tangent and the line from the point where the middle tangent touches the circle . The same will be true for the other side .. on solving we will get 70

Durvesh Kumar - 7 years, 1 month ago

same here

Irfan Shah - 7 years, 1 month ago

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