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Points $A$ and $B$ lie on a circle centered at $O$, and point $X$ is outside the circle such that $\overline{AX}$ and $\overline{BX}$ are tangent to the circle. If $\angle AXO = 26^{\circ}$, then what is the measure of minor arc $AB$, in degrees?

 Feb 9, 2020
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Points  A  and  B  divide the circle into two arcs:  major arc(AB)  and  minor arc(AB).

 

The size of angle(AXO)  =  26°.

This means that the size of angle(BXO) is also 26°.

 

The total size of angle(AXB)  =  52°.

 

Since we want to find the size of minor arc(AB), call minor arc(AB) = x.

 

Since the sum of the major arc(AB) and minor arc(AB) = 360°, major arc(AB)  =  360° - x.

 

The appropriate formula is:  [ major arc(AB) - minor arc(AB) ] / 2  =  angle(AXB).

                                                                        [  (360° - x) - x ] / 2  =  52°

                                                                                [ 360° - 2x] / 2  =  52°

                                                                                       360° - 2x  =  104°

                                                                                               256°  =  2x

                                                                                                    x  =  128°

 Feb 9, 2020

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