Triangle ABC has circumcenter O. If AB = 10 and [OAB] = 50 find the circumradius of triangle ABC.
Well, the circumcenter is where the perpendicular bisectors of the sides of the triangle meet. (right?)
If the Circumradius (x, as we shall call it) shall be Cosine (50) = 1/2 AB / X
Which is 5/Cos 50.
Which, by a calculator, is 7.77862
or 7.78
Hopefully, that's the answer.
Let $OA=OB=OC=x$. Then $x^2\sin\theta=100$ by sine area formula (I used $\angle AOB=\theta$). By Law of Cosines, $2x^2-2x^2\cos\theta=100\implies x^2(1-\cos\theta)=50$. So dividing, $\frac{1-\cos\theta}{\sin\theta}=\frac{1}{2}\implies \tan(\theta/2)=1/2$. So it's not hard to get $\sin\theta=\frac{4}{5}$ (it should remind you of the 3-4-5 triangle). Then $x^2=100*5/4=125\implies x=\sqrt{125}=\boxed{5\sqrt{5}}$
The other answer is utter nonsense.
Triangle ABC has circumcenter O. If AB = 10 and [OAB] = 50 find the circumradius of triangle ABC.
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AB = 10 [OAB] = 50
Let M be a midpoint of AB
Altitude OM = (2 * 50) / 10 = 10
OA = OB => R (Circumradius)
R = sqrt(AM2 + OM2) = √125