MathDB

20

Part of 2011 AMC 10

Problems(2)

Two chords intersection

Source: 2011 AMC A Problem 20

6/25/2011
Two points on the circumference of a circle of radius r are selected independently and at random. From each point a chord of length r is drawn in a clockwise direction. What is the probability that the two chords intersect?
<spanclass=′latex−bold′>(A)</span> 16<spanclass=′latex−bold′>(B)</span> 15<spanclass=′latex−bold′>(C)</span> 14<spanclass=′latex−bold′>(D)</span> 13<spanclass=′latex−bold′>(E)</span> 12 <span class='latex-bold'>(A)</span>\ \frac{1}{6}\qquad<span class='latex-bold'>(B)</span>\ \frac{1}{5}\qquad<span class='latex-bold'>(C)</span>\ \frac{1}{4}\qquad<span class='latex-bold'>(D)</span>\ \frac{1}{3}\qquad<span class='latex-bold'>(E)</span>\ \frac{1}{2}
probabilitygeometryperimeterAMC
Area within rhombus closest to vertex B

Source: AMC 10 2011 b Problem 20

2/24/2011
Rhombus ABCDABCD has side length 22 and ∠B=120∘\angle B = 120 ^\circ. Region RR consists of all points inside the rhombus that are closer to vertex BB than any of the other three vertices. What is the area of RR?
<spanclass=′latex−bold′>(A)</span> 33<spanclass=′latex−bold′>(B)</span> 32<spanclass=′latex−bold′>(C)</span> 233<spanclass=′latex−bold′>(D)</span> 1+33<spanclass=′latex−bold′>(E)</span> 2 <span class='latex-bold'>(A)</span>\ \frac{\sqrt{3}}{3} \qquad <span class='latex-bold'>(B)</span>\ \frac{\sqrt{3}}{2} \qquad <span class='latex-bold'>(C)</span>\ \frac{2\sqrt{3}}{3} \qquad <span class='latex-bold'>(D)</span>\ 1+\frac{\sqrt{3}}{3} \qquad <span class='latex-bold'>(E)</span>\ 2
geometryrhombustrigonometryperpendicular bisectorAMC