Math Question (Geometry).. Please Help

[Lindsey0822]

A circle is inscribed in an equilateral triangle. Radii to two of the points of tangency are shown. What is x? http://www.www.urch.com/forums/attachment.php?attachmentid=7038&stc=1

The answer given is 240 degrees. The explanation states that "the radii are two sides of a quadrilateral. A line tangent to a circle meets a radius at the point of tangency at 90 degree angle. Each angle of the equilateral triangle is 60 degrees. Therefore, the remaining angle of the quadrilateral measures 120 degrees. A circle has 360 degrees. So, the measure of angle x can be found using subtraction: 360-120=240.

I don't understand how this figure is a quadrilateral. I also don't understand how the 90 degree angle fits into the solution to the problem.

[Brent Hanneson]

Here's another approach:

Draw a 3rd line from the center to the bottom point of tangency (where the triangle side touches the circle).

This divides the circle into 3 equal sectors (pieces of pie).

360/3 = 120, so each sector has an angle of 120 degrees at the center.

So, the unlabeled angle in the diagram = 120 degrees, which makes angle x = 240 degrees.

 

Cheers,

Brent