Right triangles (the tangent ratio)

[Green1]

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?

 

[Denis]

Can you at least calculate the length of the diagonals?

 

[soroban]

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 . . .

    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: $\displaystyle \:\angle BEC\:=\:2\theta\:\approx\:28^o$