Right triangles (the tangent ratio)

Green1

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Mar 22, 2007
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I need help on this problem please help thank you



A rectangle is 80cm long and 20cm wide. Find, to the nearest degree, the acute angle formed at the intersection of the diagonals.

How do you do this problem?
 
Hello, Green1!

A rectangle is 80cm long and 20cm wide. . Find, to the nearest degree,
the acute angle formed at the intersection of the diagonals.

First, make a sketch . . .
Code:
    A *-------------------------------* B
      |   *                       *   |
      |       *               *       |
      |           *   E   * θ         |
   20 |               * - - - - - - - * F 20
      |           *       * θ         |
      |       *               *       |
      |   * θ                     *   |
    D *-------------------------------* C
                     80

We want \(\displaystyle \angle BEC.\)

Let \(\displaystyle \theta\,=\,\angle BDC\)

Note that: \(\displaystyle \,\angle BEF\,=\,\theta\,\) and \(\displaystyle \,\angle BEC\,=\,2\theta.\)

In right triangle \(\displaystyle BCD\) we have: \(\displaystyle \,\tan\theta \:=\:\frac{20}{80}\:=\:0.25\)

. . Hence: \(\displaystyle \:\theta\:=\:\tan^{-1}(0.25) \:\approx\:14.03^o\)

Therefore: \(\displaystyle \:\angle BEC\:=\:2\theta\:\approx\:28^o\)

 
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