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from Amity International School, asked a question
Subject: Math , asked on 18/1/13

If AB is a chord of a circle with centre O, AOC is diameter and AT is the tangent at the point A.

Prove that âˆ BAT=âˆ ACB

Let âˆ ACB = x and âˆ BAT = y.

A tangent makes an angle of 90 degrees with the radius of a circle,

so we know that âˆ  OAB + y = 900 â€¦â€¦..(1)
The angle in a semi-circle is 90, so âˆ  CBA = 900 .
âˆ  CBA + âˆ  OAB + âˆ ACB = 18 0 0   (Angle sum property of a triangle)
Therefore, 90 + âˆ  OAB + x = 1800

So, âˆ  OAB + x = 9 0 0â€¦â€¦â€¦â€¦.(2)
But OAB + y = 900

Therefore, âˆ  OAB + y = âˆ  OAB + x  â€¦â€¦â€¦â€¦.[From (1) and (2)]
x = y.

Hence âˆ ACB = âˆ BAT.

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