Question

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

Answer 1

The length of segment RS is:

2.4 units.

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:

`(a)/(\sin S)=(b)/(\sin R)=(c)/(\sin Q)\\\\i.e.\\\\(2.4)/(\sin S)=(3.1)/(\sin 80)=(c)/(\sin Q)`

On equating the first two equalities we have:

`(2.4)/(\sin S)=(3.1)/(\sin 80)`

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:

`(3.1)/(\sin 80)=(c)/(\sin 50.32)\\\\\\This\ implies\ that:\\\\c=2.423`

Hence, the length of line segment is:

2.4 units.

Answer 2

Explanation:

The law of sines is given by the formula;

`(a)/(\sin(A))=(b)/(\sin(B))=(c)/(\sin(C))`

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

Applying the sine rule to ΔQRS, we obtain;

`(2.4)/(\sin(S)) =(3.1)/(\sin(80\degree))`.

`2.4* \sin(80\degree) =3.1\sin(S)`.

`2.3635=3.1\sin(S)`.

`\Rightarrow (2.3635)/(3.1)=\sin(S)`.

`0.7624=\sin(S)`.

`\Rightarrow \sin^(-1)(0.7624)=S`.

`\Rightarrow 49.676=S`.

The sum of angles in a triangle is 180 degrees.

`<\:Q+49.676+80=180`

`<\:Q=180-129.676`

`\Rightarrow <\:Q=50.324`

We use the sine rule again

`(|RS|)/(\sin(50.324\degree))=(3.1)/(\sin(80\degree))`

`|RS|=(3.1)/(\sin(80\degree))* \sin(80\degree)`

`|RS|=2.423`

To the nearest tenth

`|RS|=2.4\:units`