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Length of an arc - central angle confusion

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Length of an arc - central angle confusion

by vinni.k » Sat May 18, 2013 11:21 am
Triangle ABC is inscribed in a circle, such that AC is a diameter of the circle and angle BAC is 45. If the area of triangle ABC is 84.5 square units, what is the length of arc BC ?

[spoiler](13√2π)/4[/spoiler]

I will really appreciate if anyone can explain this question. This one is from MGMAT 4th edition. The explanation says to assume central angle but the drawn diameter doesn't allow for central angle. How can central angle be drawn in this case ?

Thanks & Regards
Vinni
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by srcc25anu » Sat May 18, 2013 3:36 pm
AC is diameter. therefore Triangle CAB is right angled at B and angle B = 90 degrees. angle CAB = 45 therefore angle ACB must also be 45 degrees. So BC = AB = lets say X
Now area of ABC = 1/2 * base * Height or (1/2 * CB * BA) = 84.5
Since AB = BC therefore each of AB and BC = 1/2 * x^2 = 84.5 and x^2 = 169 or x = 13
ABC has sides in the ratio of 1:1:√2
so AC = 13√2 which is the diameter. hence radius = 13√2/2
CAB = 45 degrees so the central angle made by CB at centre O will be 90 degrees.
Arc CB = 2 * π * R * 90 / 360
Arc CB = [2 * π * (13√2 / 2)]/4 = (13√2π)/4
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