is a diameter of a circle of radius unit. is a chord perpendicular to that cuts at . If the arc is of the circumference of the circle, what is the length of the segment ?

Source: NCTM Mathematics Teacher, September 2006

**Solution
**

Circumference = . Since measure of major arc , measure of minor arc . Central angle which tends arc therefore measures radians or . Triangle is isosceles, hence altitude is also the angle bisector of central angle . Triangle is a triangle with hypotenuse equal . Thus, side has length equal .

units

**Answer**: units

**Alternative solution**

Since chord is a diameter and chord is perpendicular to , arc is a mirror image of arc in the line of reflection. Points , and divide the circumference into three equal arcs each measuring . Hence, is an equilateral triangle where the circumcenter is also the centroid of the triangle.

By the Concurrency of Medians of a Triangle theorem