Tricky Geometry Problem

[Wylie]

Problem:The television is a 52 in wide-screen television with a 9:16 ratio for its width and length.The 52 in. measurement is the diagonal measurement of the screen.There is a 2 in border on the top and bottom of the screen. The border on the side is 4 in for each side.


Question:Find the length and width of the television including the screen and its border. :?

Please help me!!!!

 

[tkhunny]

ALWAYS start with a definition. Defint what you need to know, as dictated byt he problem statement.

H = Height of screen, counting the border.
W = Width of screen, counting the border.

That seems so easy, yet it is so often overlooked. Raise this simple step on your priority lst. Trust me on this.

2" Top and Bottom

H - 4 = Height of Screen without border

4" Both Sides

W - 8 = Width of Screen without border

Ration of 9 Tall to 16 Wide

(H-4)/(W-8) = 9/16

Diagonal Measurement of 52

(H-4)^2 + (W-8)^2 = 52^2

That's about it. Two equations and two variables, you should be able to solve for both variables.

It is ALL made possible by clear definitions in the beginning. In this way, we are not confused whether we mean the measurements with or without the screen. Just look at the definitions - they're right up front!

 

[soroban]

Hello, Wylie!

The television is a 52 in wide-screen television with a 9:16 ratio for its width and length.
The 52 in. measurement is the diagonal measurement of the screen.
There is a 2 in border on the top and bottom of the screen.
The border on the side is 4 in for each side.

Find the length and width of the television including the screen and its border.

      * - - - - - - - - - - - *
      |                    *  |
      |                 *     |
      |         52   *        |
      |           *           | 9k
      |        *              |
      |     *                 |
      |  *                    |
      * - - - - - - - - - - - *
                  16k

The ratio of Width to Length is 9:16.
. . Hence, \(\displaystyle W\,=\,9k,\:L\,=\,16k\) for some constant \(\displaystyle k\).

Pythagorus tell us: \(\displaystyle \L\16k)^2\,+\,(9k)^2\:=\:52^2\;\;\Rightarrow\;\;337k^2\:=\:2704\)

. . \(\displaystyle \L k^2\:=\:\frac{2704}{337}\;\;\Rightarrow\;\;k\:=\:\frac{52}{\sqrt{337}}\:=\:\frac{52\sqrt{337}}{337}\)


Hence: \(\displaystyle \:\L\begin{array}{ccccc}L & \,=\, & 16k & \,=\, & \frac{832\sqrt{337}}{337} \\ \\ W & \,=\, & 9k & \,=\, & \frac{468\sqrt{337}}{337}\end{array}\)


For the entire television set (including the border)
. . the length is: \(\displaystyle \L\:\frac{832\sqrt{337}}{337}\,+\,8\)
. . the width is: \(\displaystyle \L\:\frac{468\sqrt{337}}{337}\,+\,4\)