Question

In circle $O$ chord $\overline{AB}$ is produced so that $\overline{BC}$ equals a radius of the circle. $\overline{CO}$ is drawn and extended to $D.$ $\overline{AO}$ is drawn. Which of the following expresses the relationship between angles $x$ and $y?$

$x=3y$

$x=2y$

$x=60^\circ$

there is no special relationship between $x$ and $y$

$x=2y$ or $x=3y,$ depending upon the length of $\overline{AB}$

Answer: A

Concepts:isosceles triangles exterior angle circle geometry

Difficulty rating: 1740

Small Hint:

Since $OB=BC,$ triangle $OBC$ is isosceles

Big Hint:

Use the exterior angle at $B$ , then use $OA=OB$

Solution:

Because $OB=BC,$ triangle $OBC$ is isosceles, so $\angle BOC=\angle BCO=y.$ Its exterior angle at $B$ is therefore $\angle ABO=2y.$ Since $OA=OB,$ triangle $AOB$ is also isosceles and $\angle OAB=2y.$ In triangle $AOC,$ the angles at $A$ and $C$ are $2y$ and $y,$ so $\angle AOC=180^\circ-3y.$ Angle $x=\angle AOD$ is supplementary to $\angle AOC,$ hence $x=3y.$

Thus, the correct answer is A.

1955 AMC 12 Problem 44

Answer: A

Difficulty rating: 1740

Small Hint:

Big Hint:

Solution:

Problem 44 in Other Years