Question : $ABC$ is a right-angled triangle with $\angle BAC=90°$ and $\angle ACB=60°$. What is the ratio of the circumradius of the triangle to the side $AB\ ?$
Option 1: $1:2$
Option 2: $1:\sqrt3$
Option 3: $2:\sqrt3$
Option 4: $2:3$
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Correct Answer: $1:\sqrt3$
Solution :
Given: $ABC$ is a right-angled triangle with $\angle BAC=90°$ and $\angle ACB=60°$.
Let the hypotenuse $BC$ of the triangle be $2r$ units.
We know,
The circumradius of a right-angled triangle is half of its hypotenuse.
So, the circumradius = $\frac{2r}{2}=r$ units
Now, $\sin60°=\frac{AB}{BC}$
⇒ $\frac{\sqrt3}{2}=\frac{AB}{BC}$
⇒ $BC:AB=2:\sqrt3$
⇒ $2r:AB=2:\sqrt3$
⇒ $r:AB=1:\sqrt3$
Hence, the ratio of the circumradius of the triangle to the side $AB$ is $1:\sqrt3$.
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