Question

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

Explanation:

Step1: Find the length of \( AC \)

In right - triangle \( ABC \), using the sine function \( \sin B=\frac{AC}{AB} \). Given \( AB = 10\mathrm{cm}\) and \( B = 30^{\circ}\), then \( AC=AB\sin B\).
Substitute \( AB = 10\) and \( \sin30^{\circ}=\frac{1}{2}\), so \( AC = 10\times\frac{1}{2}=5\mathrm{cm}\).

Step2: Find the length of \( CD \)

In right - triangle \( ACD \), using the cotangent function \( \cot D=\frac{CD}{AC}\). Also, using the tangent function \( \tan D=\frac{AC}{CD}\), then \( CD=\frac{AC}{\tan D}\).
Since \( AC = 5\mathrm{cm}\) and \( D = 25^{\circ}\), and \( \tan25^{\circ}\approx0.4663\), so \( CD=\frac{5}{\tan25^{\circ}}\approx\frac{5}{0.4663}\approx10.7\mathrm{cm}\).

Answer:

\(10.7\mathrm{cm}\) (the third option)