AMC12 2003 B

AMC12 2003 B · Q5

AMC12 2003 B · Q5. It mainly tests Ratios & proportions, Pythagorean theorem.

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?

许多电视屏幕是矩形，通常用其对角线长度来衡量。标准电视屏幕的水平长度与高度之比为 $4:3$。“$27$ 英寸”电视屏幕的水平长度最接近以下哪个数（单位：英寸）？

(A) 20 20

(B) 20.5 20.5

(D) 21.5 21.5

(E) 22 22

Answer

Correct choice: (D)

正确答案：（D）

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 $\boxed{\textbf{(D) \ } 21.5}$.

把电视屏幕分成两个直角三角形，两条直角边之比为 $4:3$，可设一条直角边为 $4x$，另一条为 $3x$，再用勾股定理。 \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*} 水平长度为 $5.4\times4=21.6$，最接近 $\boxed{\textbf{(D) \ } 21.5}$。

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