Solution to the Problem: The answer is a 56.6" traditional TV.
Let the dimensions of the HDTV be 16x by 9x.
Then using the Pythagorean Theorem, (16x)2 + (9x)2 = 602
337x2 = 3600
x = 3.2684
So, the dimensions of the TV are 52.294" by 29.416" and the area is 1,538.292 square inches.
For the traditional TV, let the dimensions be 4x by 3x.
Then 12x2 = 1,538.292
So x = 11.3221. and the dimensions are 45.2884" by 33.966".
Using the Pythagorean Theorem, the diagonal of the traditional TV is 56.611".
NOTE: But that is not the whole story.
We're all familiar with "letterboxed" TV -- bands across the top and bottom of the screen when we're watching a widescreen movie on regular TV, for instance. When that situation is reversed - when we're watching a "narrowscreen" program on HDTV - there are bands to the left and right of the picture. In other words, if we are watching anything from I Love Lucy to Seinfeld, or home videos, the pictures look like this:
AS YOU CAN SEE from the two 32-inch TVs above (the same TVs shown at the top of the page), watching "regular TV" on an HDTV with the same diagonal screen size as your old TV gives you a picture that's 33 percent smaller - only about two-thirds as big as on your old set! To avoid this pitfall, use this rule of thumb: Make sure your HDTV has a screen that's the same height as your old TV screen -- not the same area!