Question

area of a triangle: law of sines!! help!!! another tutor couldn’t do it so they referred me to someone else

Answer 1

Area of a triangle

The area of a triangle is found using its base and height:

In this case, if we rotate the given triangle, we have:

Then, we have:

`\begin{gathered} \text{area}=\frac{\text{base}\cdot\text{height}}{2} \\ \downarrow \\ \text{area}=\frac{\text{42}\cdot\text{height}}{2} \end{gathered}`

We have to find the height in order to find the area of this triangle.

Sum of inner angles

Since the sum of the inner angles of a triangle is 180º, then:

∠V + ∠X + ∠W = 180º

↓ since ∠X=55º and ∠W=67º

∠V + 55º + 67º = 180º

∠V + 122º = 180º

Solving for ∠V:

∠V + 122º = 180º

↓ taking 122º to the right side of the equation

∠V = 180º - 122º = 58º

Then,

∠V = 58º

Height

Analyzing the triangle, we can see that drawing the height a right triangle is formed:

We have that:

`\sin (67º)=\frac{\text{opposite side}}{\text{hypotenuse}}`

In this right triangle we have that:

height = opposite side

XW = hypotenuse

`\begin{gathered} \sin (67º)=\frac{\text{opposite side}}{\text{hypotenuse}} \\ \downarrow \\ \sin (67º)=\frac{\text{height}}{\text{XW}} \end{gathered}`

Then, we have an equation for the height:

`\begin{gathered} \sin (67º)=\frac{\text{height}}{\text{XW}} \\ \downarrow\text{taking XW to the left} \\ XW\cdot\sin (67º)=\text{height} \end{gathered}`

height = XW · sin(67º)

Then, if we find the length of the side XW, we will find the height.

XW measure

We have that, using the Sines Theorem (it relates the angles and their opposite side):

`\begin{gathered} (\sin(58º))/(XW)=(\sin(55º))/(VX)=(\sin(67º))/(WV) \\ \downarrow \\ (\sin(58º))/(XW)=(\sin(55º))/(VX) \\ \downarrow\text{ since VX = 42} \\ (\sin(58º))/(XW)=(\sin(55º))/(42) \end{gathered}`

Then, we have an equation for XW:

`\begin{gathered} (\sin(58º))/(XW)=(\sin(55º))/(42) \\ \end{gathered}`

Solving the equation for XW, we have:

`\begin{gathered} (\sin(58º))/(XW)=(\sin(55º))/(42) \\ \downarrow \\ \sin (58º)=(\sin(55º))/(42)XW \\ \downarrow \\ (\sin(58º))/(\sin(55º))\cdot42=XW \end{gathered}`

Solving the operation we have that:

`\begin{gathered} XW=(\sin(58º))/(\sin(55º))\cdot42\cong43.4816 \\ \end{gathered}`

Then

XW ≅ 43.4816

Height PART 2

Now, we can find the height:

height = 43.4816 · sin(67º)

height = 43.57 · 0.92 ≅ 40.025

Now, we have the height:

height ≅ 40.025

We are going to use it to find the area:

Area PART 2

`\begin{gathered} \text{area}=\frac{\text{42}\cdot\text{height}}{2} \\ \downarrow \\ \text{area}=\frac{\text{42}\cdot40.025}{2}=840.525 \end{gathered}`

Now, we have the area:

area = 840.525 m²

If we round the area to the nearest tenth (one digit after the decimal point):

840.525 ≅ 840.5

Then,

area = 840.5 m²

Final answer: area = 840.5 m²