A half-circle is inscribed within an equilateral triangle such that the diameter of the half-circle is centered on one edge of the triangle and the arc is tangent to the other two sides. What is the diameter of the semi-circle if the triangle has an edge length of 4?
+23246 Call the center of the half-circle 'O'
the right-hand base corner of the triangle 'A'
the point of tangency on the right side 'X'.
Because OX is perpendicular to AX, the angle(AXO) is a right angle.
Because the triangle is equilateral, the angle(XAO) is a 60o angle.
This means that angle(XOA) is a 30o angle.
OA has length 2, so side OX has length sqrt(3).
OA is a radius of the half-circle, so the diameter is 2·sqrt(3).
