Question

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

Answer 1

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:

Multiply both sides by 15.

`\begin{gathered} \sin C=(\sin25\degree)/(9)*15 \\ \text{Take the arcsin.} \\ C=\arcsin ((\sin25\degree)/(9)*15) \\ 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.:

Multiply both sides by sin 19.78.

`\begin{gathered} d=(9)/(\sin25)*\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} (b)/(\sin B)=(c)/(\sin C) \\ (7.207)/(\sin125.22\degree)=(c)/(\sin44.78\degree) \end{gathered}`

Multiply both sides by sin 44.78.

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