MathDB

14

Part of 2008 AMC 10

Problems(2)

Letterboxing

Source: AMC 12 2008A Problem 9

2/17/2008
Older television screens have an aspect ratio of 4:3 4: 3. That is, the ratio of the width to the height is 4:3 4: 3. The aspect ratio of many movies is not 4:3 4: 3, so they are sometimes shown on a television screen by 'letterboxing' - darkening strips of equal height at the top and bottom of the screen, as shown. Suppose a movie has an aspect ratio of 2:1 2: 1 and is shown on an older television screen with a 27 27-inch diagonal. What is the height, in inches, of each darkened strip? [asy]unitsize(1mm); defaultpen(linewidth(.8pt)); filldraw((0,0)--(21.6,0)--(21.6,2.7)--(0,2.7)--cycle,grey,black); filldraw((0,13.5)--(21.6,13.5)--(21.6,16.2)--(0,16.2)--cycle,grey,black); draw((0,2.7)--(0,13.5)); draw((21.6,2.7)--(21.6,13.5));[/asy]<spanclass=′latex−bold′>(A)</span> 2<spanclass=′latex−bold′>(B)</span> 2.25<spanclass=′latex−bold′>(C)</span> 2.5<spanclass=′latex−bold′>(D)</span> 2.7<spanclass=′latex−bold′>(E)</span> 3 <span class='latex-bold'>(A)</span>\ 2 \qquad <span class='latex-bold'>(B)</span>\ 2.25 \qquad <span class='latex-bold'>(C)</span>\ 2.5 \qquad <span class='latex-bold'>(D)</span>\ 2.7 \qquad <span class='latex-bold'>(E)</span>\ 3
ratioAMC
Rotating a Right Triangle

Source: AMC 10 2008B Problem 14

3/1/2008
Triangle OAB OAB has O \equal{} (0,0), B \equal{} (5,0), and A A in the first quadrant. In addition, \angle{ABO} \equal{} 90^\circ and \angle{AOB} \equal{} 30^\circ. Suppose that OA‾ \overline{OA} is rotated 90∘ 90^\circ counterclockwise about O O. What are the coordinates of the image of A A? (A)\ \left( \minus{} \frac {10}{3}\sqrt {3},5\right) \qquad (B)\ \left( \minus{} \frac {5}{3}\sqrt {3},5\right) \qquad (C)\ \left(\sqrt {3},5\right) \qquad (D)\ \left(\frac {5}{3}\sqrt {3},5\right) \\ (E)\ \left(\frac {10}{3}\sqrt {3},5\right)
rotationanalytic geometryLaTeXAMC