2003 AMC 10B Problem 6

P1-10~1 minPrint

Triangles: Area & Pythagorean Ratios, Percents & Averages Set up the equation/formula and compute; no special trick needed

Many television screens are rectangles that are measured by the length of their diagonals. The ratio of the horizontal length to the height in a standard television screen is $4:3$ . The horizontal length of a " $27$ -inch" television screen is closest, in inches, to which of the following?

Show solution without answering

Key insight

A screen 4 wide and 3 tall has a diagonal of 5, so the width is always 4/5 of the diagonal: 4/5 of 27 is 21.6.

Solution

1. Start with a small screen that has the right shape: $4$ wide and $3$ tall. Its diagonal is the long side of a right triangle with legs $4$ and $3$ , so by the Pythagorean theorem it is $\sqrt{16 + 9} = \sqrt{25} = 5$ .
2. So every $4:3$ screen has width, height and diagonal in the ratio $4 : 3 : 5$ . The width is $\frac{4}{5}$ of the diagonal.
3. For a $27$ -inch screen the diagonal is $27$ , so the width is
$\frac{4}{5} \times 27 = \frac{108}{5} = 21.6 \text{ inches}.$
4. Of the choices, $21.6$ is closest to $21.5$ .

The answer is $\boxed{\textbf{(D)}\ 21.5}$ .

Why this works

A rectangle with sides in the ratio $4:3$ is two $3$ - $4$ - $5$ right triangles put together, and when you scale up a shape, all its lengths grow by the same factor. Recognizing $3$ - $4$ - $5$ lets you skip solving an equation and just take a fraction of the diagonal.

Concise solution

A shorter write-up, the way contest solutions are usually written. It may take a different route from the solution above.

Write the width as $4k$ and the height as $3k$ . The diagonal splits the screen into right triangles with legs $4k$ and $3k$ , so the diagonal is $\sqrt{(4k)^2 + (3k)^2} = 5k$ : the sides and the diagonal are in the ratio $4 : 3 : 5$ .

A $27$ -inch screen has diagonal $5k = 27$ , so $k = 5.4$ and the horizontal length is $4k = 21.6$ inches.

Among the choices, $21.6$ is closest to $21.5$ , so the answer is $\boxed{\textbf{(D)}\ 21.5}$ .

Review in the AoPS books

Introduction to Geometry, Chapter 4: Perimeter and Area (§4.2)
Introduction to Algebra, Chapter 6: Ratios and Percents
Introduction to Geometry, Chapter 6: Right Triangles (§6.1, §6.2, §6.3)
Introduction to Algebra, Chapter 7: Proportion (§7.1, §7.2, §7.3)