TV aspect ratio

Question

I am working through "Geometry" by McDougal Littell. Doing self study. There is a problem that has me stumped:

A standard 27 inch analog television screen has an aspect ratio of 4:3. Use proportions and the Pythagorean theorem to calculate the length and width of the screen in inches.

I tried $x/27 = 3/4$ to come up with the width of the screen. That is: the width of the screen x, to the diagonal 27 inches as the ratio is 3 to 4. My answer was $20.25$ inches, which is wrong. The textbook gives $21.6$ inches by $16.2$ inches as the answer. I am missing something but don't know what it is.

Answer 1

Historically, all televisions were had a $4:3$ aspect ratio, hence it was only necessary to use one number to describe the size of a television. Manufacture's could have reported just the height or width, but I suspect that they chose to report the diagonal because that number is going to be larger than the other two, hence sound more impressive. In any event, when one says that a television screen is 27 inches, that means that the diagonal of the screen is 27 inches.

The question relies on the reader knowing this. As such, I think that this is a poorly worded question, and the answer in the question post (above) is certainly correct under the assumption that the 27 inches refers to the width, and not the diagonal.

The intended solution is likely something like the following:

Let $w$ and $h$ denote the width and height of the television, respectively. The diagonal of the television is $27$ inches, and so the Pythagorean theorem implies that $$ w^2 + h^2 = 27^2. \label{pythag}\tag{1}$$ The aspect ratio of the television, i.e. the ratio between width and height, is $4:3$, hence $$ \frac{w}{h} = \frac{4}{3} \implies w = \frac{4}{3} h. \label{ratio}\tag{2}$$ Substitute this into (\ref{pythag}) to get \begin{align} &27^2 = \left(\frac{4}{3} h\right)^2 + h^2 = \frac{16}{9} h^2 + h^2 = \frac{25}{9} h^2 \\ &\qquad\qquad \implies h^2 = \frac{(27\cdot 3)^2}{5^2} \\ &\qquad\qquad \implies h = \frac{81}{5} = 16.2. \end{align} Thus the height of the television is $16.2$ inches. Substituting this back into (\ref{ratio}) gives $$ w = \frac{4}{3} h = \frac{4}{3} \cdot 16.2 = 21.6. $$ Therefore the television measures $21.6$ inches by $16.2$ inches.