MathDB

19

Part of 2006 AMC 10

Problems(2)

Angles in progression.

Source: AMC 10 2006A, Problem 19

2/6/2006
How many non-similar triangle have angles whose degree measures are distinct positive integers in arithmetic progression? <spanclass=′latex−bold′>(A)</span>0<spanclass=′latex−bold′>(B)</span>1<spanclass=′latex−bold′>(C)</span>59<spanclass=′latex−bold′>(D)</span>89<spanclass=′latex−bold′>(E)</span>178 <span class='latex-bold'>(A) </span> 0 \qquad <span class='latex-bold'>(B) </span> 1 \qquad <span class='latex-bold'>(C) </span> 59 \qquad <span class='latex-bold'>(D) </span> 89 \qquad <span class='latex-bold'>(E) </span> 178
arithmetic sequenceAMC
2006 AMC 10 #19

Source:

8/20/2011
A circle of radius 2 is centered at O O. Square OABC OABC has side length 1. Sides AB‾ \overline{AB} and CB‾ \overline{CB} are extended past b b to meet the circle at D D and E E, respectively. What is the area of the shaded region in the figure, which is bounded by BD‾ \overline{BD}, BE‾ \overline{BE}, and the minor arc connecting D D and E E?
[asy] defaultpen(linewidth(0.8)); pair O=origin, A=(1,0), C=(0,1), B=(1,1), D=(1, sqrt(3)), E=(sqrt(3), 1), point=B; fill(Arc(O, 2, 0, 90)--O--cycle, mediumgray); clip(B--Arc(O, 2, 30, 60)--cycle); draw(Circle(origin, 2)); draw((-2,0)--(2,0)^^(0,-2)--(0,2)); draw(A--D^^C--E); label("AA", A, dir(point--A)); label("CC", C, dir(point--C)); label("OO", O, dir(point--O)); label("DD", D, dir(point--D)); label("EE", E, dir(point--E)); label("BB", B, SW);[/asy]
(A) \frac {\pi}3 \plus{} 1 \minus{} \sqrt {3} \qquad (B) \frac {\pi}2\left( 2 \minus{} \sqrt {3}\right) \qquad (C) \pi\left(2 \minus{} \sqrt {3}\right) \qquad (D) \frac {\pi}{6} \plus{} \frac {\sqrt {3} \minus{} 1}{2} \\ \qquad \indent (E) \frac {\pi}{3} \minus{} 1 \plus{} \sqrt {3}
geometrytrigonometryPythagorean Theorem