Question

the diagonal of a rectangle is 321 millimeters, while the longer side is 244 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: mm angle formed from short side and diagonal: angle formed from long side and diagonal:

Explanation:

Step1: Find the shorter side using Pythagorean theorem

Let the diagonal be $d = 321$ mm, the longer side be $l=244$ mm and the shorter side be $s$. According to the Pythagorean theorem $s=\sqrt{d^{2}-l^{2}}$. So $s=\sqrt{321^{2}-244^{2}}=\sqrt{(321 + 244)(321-244)}=\sqrt{565\times77}=\sqrt{43505}\approx209$ mm.

Step2: Find the angle between the short - side and the diagonal

Let $\theta_1$ be the angle between the short - side and the diagonal. We know that $\cos\theta_1=\frac{s}{d}$. So $\theta_1=\cos^{-1}(\frac{209}{321})\approx49^{\circ}$.

Step3: Find the angle between the long - side and the diagonal

Let $\theta_2$ be the angle between the long - side and the diagonal. Since the two angles between the sides and the diagonal of a rectangle are complementary, $\theta_2 = 90^{\circ}-\theta_1$. So $\theta_2=90 - 49=41^{\circ}$.

Answer:

Shorter side: 209 mm
Angle formed from short side and diagonal: 49°
Angle formed from long side and diagonal: 41°