Question

find th acute angle between the diagonal andthe longer side of a rectangle with sides 12 cm and 5 cm

Answer 1

Answer on Question#38714 - Math - Geometry

Question: Find the acute angle between the diagonal and the longer side of a rectangle with sides 12 cm and 5 cm.

Solution. Let us begin by making a drawing:

Here we have denoted the vertices of the rectangle as $A, B, C, D$ , supposing that $AD = 12$ cm and $CD = 5$ cm.

The sought angle between the diagonal and the longer side of $ABCD$ is $\angle CAD$ .

Note that the triangle $ADC$ is a right triangle (since $\angle ADC$ is an angle of a rectangle and thus $\angle ADC = 90{}^\circ$ ). In this triangle, we are given the lengths of adjacent and opposite sides to angle $\angle CAD$ ; thus, we can find the tangent of this angle using the standard formula:

$$\tan \angle C A D = \frac {C D}{A D} = \frac {5}{1 2}.$$

The angle $\angle CAD$ itself can be found using the inverse function:

$$\angle C A D = \tan^ {- 1} \left(\frac {5}{1 2}\right) = 2 2. 6 2 {}^ {\circ}.$$

Answer. The acute angle between the diagonal and the longer side of a rectangle with sides 12 cm and 5 cm equals $22.62{}^{\circ}$ .