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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?

 

 Jul 1, 2020
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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).

 Jul 1, 2020

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