Q:

Find the value of x in the figure belowA) 94B) 78C) 110°D) 210

Accepted Solution

A:
There's the tangent secant angle theorem which says the measure of angle B is half the difference of the two intercepted arcs.  That means58 = (1/2) (210 - x)116 = 210 - xx = 94Answer: AThat's not a very intuitive answer.  Let's see if we can see why.  Let's call the center of the circle O and the unlabeled intersection at the top D.Angle DOC=210 degrees the long way, that's what an arc measure meansAngle DOC=360-210=150 degrees the short wayWe have angle BCO=90 degrees because it's a tangent.So angle BDO is the fourth angle in a quadrailateral BCDOBDO = 360 - DBC - BCO - DOC = 360 - (58 + 150+90) = 62 degreesADO is isosceles, with two radii for sides.  So DAO=62 degreesThat leaves DOA = 180 - 62 - 62 = 56 degreesAngle AOC = xx + arc AD = DOCx + 56 = 150x = 94That checks.