1955 AMC 12 Problem 44

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}

Since

OB=BC,

triangle

OBC

is isosceles

Use the exterior angle at

B

, then use

OA=OB

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.

Problem 44 in other years