Question

find the value of x. give your answers to 3 significant figures.

Answer 1

Answer:

x=6.21 cm

Explanation:

In the given triangle, we are required to find the length of AB, that is x cm.

First, find the value of angle ACD.

Using the Law of Sines:

$\begin{gathered} \frac{\sin C}{c}=\frac{\sin A}{a} \\ \implies\frac{\sin C}{15}=\frac{\sin25\degree}{9}\text{ where }\begin{cases}c=AD=15\operatorname{cm} \\ a=CD=9\operatorname{cm} \\ A=25\degree\end{cases} \end{gathered}$

Multiply both sides by 15.

$\begin{gathered} \sin C=\frac{\sin25\degree}{9}\times15 \\ \text{Take the arcsin.} \\ C=\arcsin (\frac{\sin25\degree}{9}\times15) \\ C=44.778\degree \end{gathered}$

Since angle C in the figure is an obtuse angle:

$\begin{gathered} m\angle\text{ACD}=180\degree-44.778\degree \\ m\angle\text{ACD}=135.22\degree \end{gathered}$

Next, we find the value of angle ADC.

In triangle ACD:

$\begin{gathered} m\angle A+m\angle C+m\angle D=180\degree\text{ (Sum of angles in a triangle)} \\ 25\degree+135.22\degree+m\angle D=180\degree \\ m\angle D=180\degree-25\degree-135.22\degree \\ \implies m\angle ADC=19.78\degree \end{gathered}$

The diagram below shows the two angles.

The next step is to find the length of AC using the Law of Sines.:

$\begin{gathered} \frac{d}{\sin D}=\frac{a}{\sin A} \\ \implies\frac{d}{\sin 19.78\degree}=\frac{9}{\sin25}\text{ where }\begin{cases}D=19.78\degree \\ a=CD=9\operatorname{cm} \\ A=25\degree\end{cases} \end{gathered}$

Multiply both sides by sin 19.78.

$\begin{gathered} d=\frac{9}{\sin25}\times\sin 19.78\degree \\ d=14.9999 \\ d\approx7.207 \\ AC\approx7.207 \end{gathered}$

Find the value of angle ACB in triangle ABC.

The sum of angles on a straight line is 180 degrees.

$\begin{gathered} 135.22\degree+m\angle\text{ACB}=180\degree \\ m\angle\text{ACB}=180\degree-135.22\degree \\ m\angle\text{ACB}=44.78\degree \end{gathered}$

Find the value of angle ABC in triangle ABC.

$m\angle\text{ABC}=180-(10+44.78)=125.22\degree$

From the diagram below:

Applying the Law of Sines to triangle ABC:

$\begin{gathered} \frac{b}{\sin B}=\frac{c}{\sin C} \\ \frac{7.207}{\sin125.22\degree}=\frac{c}{\sin44.78\degree} \end{gathered}$

Multiply both sides by sin 44.78.

$\begin{gathered} c=\frac{7.207}{\sin125.22\degree}\times\sin 44.78\degree \\ c=6.21\operatorname{cm} \end{gathered}$

The length of x (i.e AB) is 6.21 cm (correct to 3 significant figures).