Question

Find the area of the triangle given: b=6ft A=20° C=110°

Answer 1

Let's take a look at our triangle:

Now, using the law of sines, we'll get that:

`\begin{gathered} (b)/(\sin B)=(c)/(\sin C) \\ \text{and} \\ (b)/(\sin B)=(a)/(\sin A) \end{gathered}`

We know that the sum of the three interior angles of a trianlge is 180°. Therefore, we can conclude that:

`\angle B=50`

Now. solving for a and c,

`\begin{gathered} (b)/(\sin B)=(c)/(\sin C)\rightarrow c=(b\sin C)/(\sin B)\Rightarrow c=7.36 \\ \text{and} \\ (b)/(\sin B)=(a)/(\sin A)\rightarrow a=(b\sin A)/(\sin B)\Rightarrow a=2.68 \end{gathered}`

This way, we know the lenght of the three sides of the triangle:

`\begin{gathered} a=2.68 \\ b=6 \\ c=7.36 \end{gathered}`

Now, we can calculate the area of such triangle using Heron's formula:

`A=\sqrt[]{s(s-a)(s-b)(s-c)}`

Where:

`s=(a+b+c)/(2)`

Using this, and the sides we've just calculated, we'll have that:

`s=(2.68+6+7.36)/(2)\Rightarrow s=8.02`

Thereby,

`\begin{gathered} A=\sqrt[]{8.02(8.02-2.68)(8.02-6)(8.02-7.36)} \\ \Rightarrow A=7.56 \end{gathered}`

We can conclude that the area of the triangle is 7.56 square feet