# Problem of the Day

The aspect ratio of an image is the ratio of its width to its height. If the aspect ratio of an image on a particular television is 16:9, and the diagonal of the image measures inches, which of the following is an expression for the height of the image, in terms of ?

A. B. C. D. E. [latexpage]

D. $\displaystyle \frac{9x\sqrt{337}}{337}$

See the Solution

### Solution

[latexpage]

The diagonal of the image forms a right triangle with the height and width. Use the Pythagorean Theorem to find the length of the diagonal with a width and height of 9 and 16:

$d^2=9^2+16^2$
$d^2=337$
$d=\sqrt{337}$

This tells us that the ratio of the lengths of the width, height, and diagonal, is $\displaystyle 16:9:\sqrt{337}$. Let $x$ represent the measure of the diagonal, and $h$ represent the measure of the height. From the ratio that we established, we can say that $\displaystyle \frac{h}{x}=\frac{9}{\sqrt{337}}$. Solving for $h$ and simplifying:

$\displaystyle \frac{h}{x}=\frac{9}{\sqrt{337}}$

$h=\displaystyle \frac{9x}{\sqrt{337}}$

$h=\displaystyle \frac{9x}{\sqrt{337}}\cdot \frac{\sqrt{337}}{\sqrt{337}}$

$h=\displaystyle \frac{9x\sqrt{337}}{337}$