1969 AMC 12 Problem 8

Attempt Problem 8 of the 1969 AMC 12 below, then check your answer against the professionally curated solution from LIVE by Po-Shen Loh. You can also try the full timed exam, view all 1969 AMC 12 solutions, or check the answer key.

All problems are used with official legal permission of the Mathematical Association of America (MAA).

Triangle

ABC

is inscribed in a circle. The measures of the non-overlapping minor arcs

AB,

BC,

and

CA

are, respectively,

x+75^\circ,

2x+25^\circ,

3x-22^\circ.

Then one interior angle of the triangle, in degrees, is:

57\dfrac12

59

60

61

122

The three non-overlapping arcs sum to

360^\circ

Each inscribed angle is half the measure of its intercepted arc

The arc sum gives

\begin{aligned} 6x+78&=360,\\ x&=47. \end{aligned}

The arcs then measure

122^\circ,

119^\circ,

and

119^\circ.

The angle intercepting the

122^\circ

arc measures

61^\circ.

Therefore, the correct answer is D.

Problem 8 in other years