MathDB

22

Part of 2017 AMC 10

Problems(2)

Triangular tangents

Source: 2017 AMC10 #22

2/8/2017
Sides AB‾\overline{AB} and AC‾\overline{AC} of equilateral triangle ABCABC are tangent to a circle at points BB and CC respectively. What fraction of the area of △ABC\triangle ABC lies outside the circle?
<spanclass=′latex−bold′>(A)</span>43π27−13<spanclass=′latex−bold′>(B)</span>32−π8<spanclass=′latex−bold′>(C)</span>12<spanclass=′latex−bold′>(D)</span>3−23π9<spanclass=′latex−bold′>(E)</span>43−43π27 <span class='latex-bold'>(A) </span>\dfrac{4\sqrt{3}\pi}{27}-\frac{1}{3}\qquad <span class='latex-bold'>(B) </span> \frac{\sqrt{3}}{2}-\frac{\pi}{8}\qquad <span class='latex-bold'>(C) </span> \frac{1}{2} \qquad <span class='latex-bold'>(D) </span>\sqrt{3}-\frac{2\sqrt{3}\pi}{9}\qquad <span class='latex-bold'>(E) </span> \frac{4}{3}-\dfrac{4\sqrt{3}\pi}{27}
AMCAMC 10geometry2017 AMC 10A
Crazy Coordinate Crisis

Source: 2017 AMC 10B #22 / AMC 12B #18

2/16/2017
The diameter AB‾\overline{AB} of a circle of radius 22 is extended to a point DD outside the circle so that BD=3BD=3. Point EE is chosen so that ED=5ED=5 and the line EDED is perpendicular to the line ADAD. Segment AE‾\overline{AE} intersects the circle at point CC between AA and EE. What is the area of △ABC\triangle ABC?
<spanclass=′latex−bold′>(A) </span>12037<spanclass=′latex−bold′>(B) </span>14039<spanclass=′latex−bold′>(C) </span>14539<spanclass=′latex−bold′>(D) </span>14037<spanclass=′latex−bold′>(E) </span>12031<span class='latex-bold'>(A) \ </span> \frac{120}{37}\qquad <span class='latex-bold'>(B) \ </span> \frac{140}{39}\qquad <span class='latex-bold'>(C) \ </span> \frac{145}{39}\qquad <span class='latex-bold'>(D) \ </span> \frac{140}{37}\qquad <span class='latex-bold'>(E) \ </span> \frac{120}{31}
analytic geometryAMCAMC 10geometry2017 AMC 10B2017 AMC 12BAMC 12