Circles
Circumcircle of Triangle
Grade 11
Question:
<p>An altitude BD and a bisector BE are drawn in the triangle ABC from the vertex B. It is known that the length of side AC = 1, and the magnitudes of the angles \(\angle BEC\), \(\angle ABD\), \(\angle ABE\), \(\angle BAC\) form an arithmetic progression.</p><p>The area of circle circumscribing \(\triangle ABC\) is:</p>
<p>(a) \(\frac{\pi}{8}\)</p>
<p>(b) \(\frac{\pi}{4}\)</p>
<p>(c) \(\frac{\pi}{2}\)</p>
<p>(d) \(\pi\)</p>
Step-by-Step Solution
Key Concept: Use the arithmetic progression condition on the angles along with properties of altitude and angle bisector to determine the triangle's angles and apply the circumradius formula.
<p>From the given conditions about the arithmetic progression of angles and the properties of altitude and angle bisector from vertex B, combined with AC = 1, we can determine the triangle's dimensions.</p><p>Using the circumradius formula \(R = \frac{a}{2\sin A}\) and the derived angle measures, the circumradius can be calculated.</p><p>Area of circumscribed circle = \(\pi R^2 = \frac{\pi}{4}\)</p><p>∴ Answer is (b).</p>
Correct Answer: b