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2008 AMC 10A Problem 14

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2008 AMC 10A Problem 14AMC 10 Step by Step
Name:                             Date: September 13, 2026
Triangles: Area & Pythagorean · difficulty 2 of 5 · about 3 min · position P11-15

Older television screens have an aspect ratio of 4:34: 3 . That is, the ratio of the width to the height is 4:34: 3 . The aspect ratio of many movies is not 4:34: 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 2:12: 1 and is shown on an older television screen with a 2727 -inch diagonal. What is the height, in inches, of each darkened strip? figure

  1. A)

    22

  2. B)

    2.252.25

  3. C)

    2.52.5

  4. D)

    2.72.7

  5. E)

    33

Answer

D

Key insight

A 4:3 screen with diagonal 27 is a scaled 3-4-5 triangle, 21.6 by 16.2; a 2:1 movie at full width is 10.8 tall, leaving 5.4 to split.

Solution

A screen with width-to-height ratio 4:34:3 has width 4k4k, height 3k3k and diagonal 5k5k (a 33-44-55 triangle). The diagonal is 2727, so k=5.4k = 5.4: the screen is 21.621.6 inches wide and 16.216.2 inches tall.

The movie fills the full width 21.621.6 and has aspect ratio 2:12:1, so its height is 21.62=10.8\frac{21.6}{2} = 10.8 inches.

The leftover height is 16.2−10.8=5.416.2 - 10.8 = 5.4 inches, shared equally by the top and bottom strips:
5.42=2.7.\frac{5.4}{2} = 2.7.

The answer is (D) 2.7\boxed{\textbf{(D)}\ 2.7}.

Why this works

"Aspect ratio plus diagonal" is a Pythagorean setup, and 4:34:3 is the classic 33-44-55 triple, so the diagonal converts directly to actual dimensions. After that it is a matter of tracking which dimension the movie shares with the screen (the width) and splitting the unused height.

Alternative approach

Work in units of kk: the movie height is 4k2=2k\frac{4k}{2} = 2k, so the two strips together are 3k−2k=k3k - 2k = k and each strip is k2=2710=2.7\frac{k}{2} = \frac{27}{10} = 2.7.

The trap

Reporting the combined dark height 5.4 or the movie height 10.8 instead of the height of one strip.

Common mistakes

  • Reporting the combined dark height 5.4 or the movie height 10.8 instead of the height of one strip.
  • Fitting the movie to the screen's height instead of its width, which would make the movie wider than the screen.

Techniques

Set up the equation/formula and compute; no special trick needed