Circles
Inscribed and Central Angles
Grade 11
Question:
<p>Let 'O' be the circumcentre of \(\triangle ABC\). Under the same conditions as Problem 1, the radius of circle inscribed in \(\triangle BOC\) is:</p>
<p>(a) \(\frac{1}{8\sqrt{3}}\)</p>
<p>(b) \(\frac{1}{4\sqrt{3}}\)</p>
<p>(c) \(\frac{1}{2\sqrt{3}}\)</p>
<p>(d) \(\frac{1}{2}\)</p>
Step-by-Step Solution
Key Concept: Use the relationship between circumcentre O and the inscribed angle to find angle BOC, then apply the inradius formula to triangle BOC.
<p><strong>Solution:</strong> From Problem 1, we determined the angles of \(\triangle ABC\). The circumcentre O and the triangle BOC can be analyzed using the properties derived. The angle \(\angle BOC = 2\angle BAC\) (central angle is twice the inscribed angle). Using the inradius formula \(r = \frac{\text{Area}}{s}\) where s is the semi-perimeter of \(\triangle BOC\), and with circumradius \(R = \frac{1}{2}\), we calculate the inradius of \(\triangle BOC = \frac{1}{8\sqrt{3}}\).</p>
Correct Answer: a