Question

find the length of \\(\overline{ac}\\). use that length to find the length of \\(\overline{cd}\\). what is the length of \\(\overline{cd}\\)? round to the nearest tenth. \\(\bigcirc\\) 2.3 cm \\(\bigcirc\\) 4.0 cm \\(\bigcirc\\) 10.7 cm \\(\bigcirc\\) 18.6 cm

Explanation:

Step1: Find length of \( \overline{AC} \)

In right triangle \( ABC \), \( \angle B = 30^\circ \), \( AB = 10 \) cm. Using sine function: \( \sin(30^\circ)=\frac{AC}{AB} \). So \( AC = AB \times \sin(30^\circ)=10\times\frac{1}{2}=5 \) cm.

Step2: Find length of \( \overline{CD} \)

In right triangle \( ACD \), \( \angle D = 25^\circ \), \( AC = 5 \) cm. Using cotangent function: \( \cot(25^\circ)=\frac{CD}{AC} \), so \( CD = AC \times \cot(25^\circ) \). \( \cot(25^\circ)=\frac{\cos(25^\circ)}{\sin(25^\circ)}\approx\frac{0.9063}{0.4226}\approx2.145 \). Then \( CD\approx5\times2.145 = 10.725\approx10.7 \) cm.

Answer:

10.7 cm (corresponding to the option "10.7 cm")