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?

Guest Jul 1, 2020

#1**0 **

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 60^{o} angle.

This means that angle(XOA) is a 30^{o} 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).

geno3141 Jul 1, 2020