Chapter 8.5, Problem 24WE

To determine

To find: the acute angle formed at the intersection of the diagonals.

Answer to Problem 24WE

The acute angle $\angle DOC$ formed at the intersection of the diagonals is $28°$ .

Explanation of Solution

Given:

The length and width of the rectangle are 80 cm and 20 cm respectively.

Calculation:

Diagonals of the rectangle are bisector. Draw a perpendicular from the intersection point to the base of the rectangle.

Hence, the rectangle is shown below:

In triangle OEC,

  $\begin{array}{c}\tan \angle COE=\frac{\text{Height}}{\text{Base}}\\ \tan \angle COE=\frac{40}{10}\\ \angle COE\approx 76°\end{array}$

Thus, the angle BOC is calculated as follows:

  $\begin{array}{c}\angle BOC=2\angle COE\\ =2\left(76°\right)\\ =152°\end{array}$

The acute angle $\angle DOC$ formed at the intersection of the diagonals is calculated as follows:

  $\begin{array}{c}\angle DOC=180°-\angle BOC\\ =180°-152°\\ =28°\end{array}$

Thus, the acute angle $\angle DOC$ formed at the intersection of the diagonals is $28°$ .