Question

What is the value of a + b + c? You may assume that the ray is tangent to the circle ?

Answer 1

PICTURE

Answer 2

What I can gather so far...

The X-ed lines are both radii of the circle, which means they must be equivalent and the associated triangle is isosceles. This means the indicated angle (opposite \(a\), labeled with an arrow) is the same as \(a\).

The arc labeled 56 corresponds to its sector of angle 56 degrees, and this angle is supplementary to the remaining angle in the triangle, which is \(180-54=126^\circ\). Hence \(a+a+126=180^\circ~~\iff~~a=27^\circ\).

Similarly, you can determine \(c\) by finding the remaining arc length. The chord running through the center is the diameter, so the arc to the right (moving counterclockwise) has measure 94, so the angle of the triangle (between the dashed line and the radius perpendicular to the tangent ray) is also \(94^\circ\). This new triangle is also isosceles, so \(c+c+94=180^\circ~~\iff~~c=43^\circ\).

Not sure about \(b\) just yet...