Question

the diagonal of a rectangle is 480 millimeters, while the longer side is 371 millimeters. find the shorter side of the rectangle and the angles the diagonal makes with each side. round to the nearest whole number. shorter side: 305 mm angle formed from short side and diagonal: angle formed from long side and diagonal:

Explanation:

Step1: Use Pythagorean theorem to find shorter side

Let the diagonal be $c = 480$ mm, the longer side be $a = 371$ mm, and the shorter side be $b$. By the Pythagorean theorem $b=\sqrt{c^{2}-a^{2}}=\sqrt{480^{2}-371^{2}}=\sqrt{(480 + 371)(480 - 371)}=\sqrt{851\times109}=\sqrt{92759}\approx305$ mm.

Step2: Find angle between short - side and diagonal

Let $\theta_1$ be the angle between the short - side and the diagonal. We know that $\cos\theta_1=\frac{b}{c}$. So $\theta_1=\arccos(\frac{305}{480})\approx50^{\circ}$

Step3: Find angle between long - side and diagonal

Let $\theta_2$ be the angle between the long - side and the diagonal. We know that $\cos\theta_2=\frac{a}{c}$. So $\theta_2=\arccos(\frac{371}{480})\approx40^{\circ}$

Answer:

Angle formed from short side and diagonal: $50$
Angle formed from long side and diagonal: $40$