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Find the circumradius of triangle $ABC.$

[asy] unitsize(1 cm);pair A, B, C;A = (1,3);B = (0,0);C = (4,0);draw(A--B--C--cycle);label(

Guest Apr 12, 2022

+64

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Mhmmmmmm

Kakashi Apr 12, 2022

CPhill · Answer 1 · 2022-04-12T03:50:24

C = 45°

By the Law of Sines

8 /sin 45 = BC / sin 60

8 sin 60 / sin 45 = BC = 4sqrt 6

And

8 /sin 45 = AC /sin75

8sin75 / sin 45 = AC = 4 ( 1 + sqrt 3)

Area of triangle = A = (1/2)AB *BC *sin75 = (1/2) ( 8) ( 4sqrt 6) sin 75 = 24 + 8sqrt (3)

AB * BC * AC = 8 * [ 4sqrt (6) ] * [ 4 (1 + sqrt (3) ) ] = 128 sqrt (6) ( 1 + sqrt (3) )

Circumradius = [ AB * BC * AC ] / [ 4 A ] =

[ 128 sqrt (6) ( 1 + sqrt (3) ) ] / [ 4 * ( 24 + 8sqrt (3)) ] = 4sqrt (2)