AMC 10 · 2008 · #14

Grade 8 geometry-2d
Archetype Coordinate Plane Figure Computation └ Sub-archetype Pythagorean / Distance Formula for a Length
ratio-proportionpythagorean-theoremarea-rectangles convert-to-algebraidentify-subproblems ↑ Prerequisites: pythagorean-theoremratio-proportion
📏 Medium solution 💡 2 insights 📊 Diagram
Problem
A television screen has width-to-height ratio 4:3 and a 27-inch diagonal. A movie with width-to-height ratio 2:1 is shown on it so the movie fills the full width, leaving two dark strips of equal height (top and bottom). Find the height, in inches, of one strip.

Pick an answer.

(A)
$\ 2$
(B)
$\ 2.25$
(C)
$\ 2.5$
(D)
$\ 2.7$
(E)
$\ 3$

AMC 10 2008 problem © Mathematical Association of America (MAA AMC). Reproduced for educational use.

How to solve
Strategy Introduce a Variable

The ratios 4:3 and 2:1 fix the shapes but not the actual sizes, so Tool #4 (Introduce a Variable) names one scale factor k and writes every length in terms of it. The picture (Tool #1) shows a right triangle made by the width, height, and diagonal, which turns the single number 27 into an equation for k. Then Tool #7 (Identify Subproblems) breaks the goal into a chain: find the screen size, find the movie height, then find the leftover and halve it.

1STEP 1

Name the screen size with one variable

Ratio 4:3 means the screen is 4k wide and 3k tall for one positive scale factor k.

width = 4k, height = 3k
2STEP 2

Use the diagonal to solve for k

Pythagoras gives (4k)² + (3k)² = 25k², so the diagonal is 5k. Then 5k = 27 yields k = 5.4.

(4k)² + (3k)² = 25k² → 5k = 27 → k = 5.4
3STEP 3

Find the screen's width and height

Substituting k = 5.4 makes the screen 21.6 inches wide and 16.2 inches tall.

width = 21.6, height = 16.2
4STEP 4

Find the movie's height

The movie is 21.6 wide too, and a 2:1 shape is half as tall as it is wide, so it stands 10.8 inches.

movie height = 21.6/2 = 10.8
5STEP 5

Take the leftover and split it in two

Dark height totals 16.2 - 10.8 = 5.4 inches, split evenly by the two strips, so each is 2.7 inches — answer (D).

one strip = (16.2 - 10.8)/2 = 5.4/2 = 2.7 → (D)
Answer
2.7
The strip height 2.7 is small next to the 16.2-inch screen, which fits two thin bands framing a tall movie. Checking the total: two strips of 2.7 plus the movie's 10.8 gives 2.7 + 10.8 + 2.7 = 16.2, exactly the screen height. The figure's strips run from 0 to 2.7 and from 13.5 to 16.2, each 2.7 tall, confirming (D).
💡Key takeaway

A 4:3 screen is a scaled 3-4-5 triangle, so a 27-inch diagonal makes it 16.2 tall; the 10.8-tall movie leaves 5.4 of darkness split into two 2.7-inch strips.

  • Name the screen size with one variable
  • Use the diagonal to solve for k
  • Find the screen's width and height
  • Find the movie's height
  • Take the leftover and split it in two

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