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What is the length of line segment RS? Use the law of sines to find the answer. Round to the nearest tenth.

What is the length of line segment RS Use the law of sines to find the answer Round to the nearest tenth class=

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Answer:


Step-by-step explanation:


The law of sines is given by the formula;

[tex]\frac{a}{\sin(A)}=\frac{b}{\sin(B)}=\frac{c}{\sin(C)}[/tex]

where A,B,C are the interior angles of triangle ABC and [tex]a,b,c[/tex] are the sides opposite these angles.


Applying the sine rule to ΔQRS, we obtain;

[tex]\frac{2.4}{\sin(S)} =\frac{3.1}{\sin(80\degree)}[/tex].


[tex]2.4\times \sin(80\degree) =3.1\sin(S)[/tex].


[tex]2.3635=3.1\sin(S)[/tex].


[tex]\Rightarrow \frac{2.3635}{3.1}=\sin(S)[/tex].


[tex]0.7624=\sin(S)[/tex].



[tex]\Rightarrow \sin^{-1}(0.7624)=S[/tex].


[tex]\Rightarrow 49.676=S[/tex].


The sum of angles in a triangle is 180 degrees.

[tex]<\:Q+49.676+80=180[/tex]


[tex]<\:Q=180-129.676[/tex]


[tex]\Rightarrow <\:Q=50.324[/tex]



We use the sine rule again


[tex]\frac{|RS|}{\sin(50.324\degree)}=\frac{3.1}{\sin(80\degree)}[/tex]


[tex]|RS|=\frac{3.1}{\sin(80\degree)}\times \sin(80\degree)[/tex]


[tex]|RS|=2.423[/tex]


To the nearest tenth


[tex]|RS|=2.4\:units[/tex]

Answer:

The length of segment RS is:

                              2.4 units.

Step-by-step explanation:

We will use the Law of Cosine to find the length of segment RS.

Let the  segment RS be denoted by 'c'.

Let a=RQ=2.4 units.

b=QS=3.1 units.

Now, according to the law of cosine we have:

[tex]\dfrac{a}{\sin S}=\dfrac{b}{\sin R}=\dfrac{c}{\sin Q}\\\\i.e.\\\\\dfrac{2.4}{\sin S}=\dfrac{3.1}{\sin 80}=\dfrac{c}{\sin Q}[/tex]

On equating the first two equalities we have:

[tex]\dfrac{2.4}{\sin S}=\dfrac{3.1}{\sin 80}[/tex]

Hence, on solving we get:

S=49.68°

Now as we know that in a triangle the sum of all the angles of a triangle is:

180 degree.

Hence,

∠Q+∠R+∠S=180°

⇒   ∠Q+80°+49.68°=180°

⇒  ∠Q=50.32°

Similarly, on equating the second and third equality we have:

[tex]\dfrac{3.1}{\sin 80}=\dfrac{c}{\sin 50.32}\\\\\\This\ implies\ that:\\\\c=2.423[/tex]

          Hence, the length of line segment is:

                             2.4 units.

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