Question

find the length of \\(\overline{ac}\\). use that length to find the length of \\(\overline{cd}\\)? round to the nearest tenth.
\\(\overline{cd}\\).
\\(\boldsymbol{2.3}\\) cm
\\(\boldsymbol{4.0}\\) cm
\\(\boldsymbol{10.7}\\) cm
\\(\boldsymbol{18.6}\\) cm
(image of a geometric figure with right angle at c, triangle abc with ab = 10 cm, angle at b is 30°, triangle adc with angle at d is 25°)

Explanation:

Step1: Find length of AC

In right triangle \( ABC \), \( \angle B = 30^\circ \), hypotenuse \( AB = 10 \) cm. Using \( \sin(30^\circ)=\frac{AC}{AB} \), so \( AC = AB \times \sin(30^\circ) = 10 \times 0.5 = 5 \) cm.

Step2: Find length of CD

In right triangle \( ACD \), \( \angle D = 25^\circ \), opposite side \( AC = 5 \) cm. Using \( \tan(25^\circ)=\frac{AC}{CD} \), so \( CD=\frac{AC}{\tan(25^\circ)}=\frac{5}{\tan(25^\circ)} \). Calculate \( \tan(25^\circ)\approx0.4663 \), then \( CD\approx\frac{5}{0.4663}\approx10.7 \) cm.

Answer:

10.7 cm (corresponding to the option with 10.7 cm)