2003 AMC 10B Problems/Problem 6

Problem

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?

$\textbf{(A) } 20 \qquad\textbf{(B) } 20.5 \qquad\textbf{(C) } 21 \qquad\textbf{(D) } 21.5 \qquad\textbf{(E) } 22$

Solution

If you divide the television screen into two right triangles, the legs are in the ratio of $4 : 3$ , and we can let one leg be $4x$ and the other be $3x$ . Then we can use the Pythagorean Theorem.

$\begin{align*}(4x)^2+(3x)^2&=27^2\\ 16x^2+9x^2&=729\\ 25x^2&=729\\ x^2&=\frac{729}{25}\\ x&=\frac{27}{5}\\ x&=5.4\end{align*}$

The horizontal length is $5.4\times4=21.6$ , which is closest to $\boxed{\mathrm{(D) \ } 21.5}$ .

2003 AMC 10B (Problems • Answer Key • Resources)
Preceded by Problem 5	Followed by Problem 7
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2003 AMC 10B Problems/Problem 6

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