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
(image: triangle b d with a right angle at c, segment a c is perpendicular to b d. side a b is 10 cm, angle at b is 30°, angle at d is 25°.)

Explanation:

Step1: Find length of $\overline{AC}$

In right $\triangle ABC$, use $\sin\theta=\frac{\text{opposite}}{\text{hypotenuse}}$.
$\sin(30^\circ)=\frac{AC}{10}$
$AC = 10\times\sin(30^\circ)=10\times0.5=5$ cm

Step2: Find length of $\overline{CD}$

In right $\triangle ACD$, use $\tan\theta=\frac{\text{opposite}}{\text{adjacent}}$.
$\tan(25^\circ)=\frac{AC}{CD}$
$CD=\frac{AC}{\tan(25^\circ)}=\frac{5}{\tan(25^\circ)}$
Calculate $\tan(25^\circ)\approx0.4663$, so $CD\approx\frac{5}{0.4663}\approx10.7$ cm

Answer:

10.7 cm (Option C)