Question

a surveyor locating the corners of a four-sided of property started at one corner and walk 200 feet in the direction of N80°E 5o reach the next corner he turned and walked to north 160 feet to the next corner of the property he did turn and walk due west to get to the 4th corner of the property finally he walked in the direction S15°E to get back to the starting point. What is the area of the property is in square feet?

Answer 1

we have three figures, we must find the area of ​​each one and at the end, add them

lower triangle

we must find x and y to calculate the area, we will use trigonometric ratios

$\begin{gathered} \sin (80)=\frac{x}{200} \\ \\ x=200\sin (80) \\ x=197 \end{gathered}$$\begin{gathered} \sin (10)=\frac{y}{200} \\ \\ y=200\sin (10) \\ y=34.73 \end{gathered}$

now calculate the area

$\begin{gathered} A_{T1}=\frac{b\times h}{2} \\ \\ A_{T1}=\frac{y\times x}{2}=\frac{34.73\times197}{2} \\ \\ A_{T1}_{}=3420.9 \end{gathered}$

the area of the triangle is 3420.9 square feet

Rectangle

we have the height (160ft) and the base we calculate it in the previous step (x=197ft)

the area is

$\begin{gathered} A_R=b\times h \\ A_R=197\times160 \\ A_R=31520 \end{gathered}$

the area of the rectangle is 31520 square feet

Left Triangle

we must use trigonometric ratios to calculate Z

$\begin{gathered} \tan (15)=\frac{Z}{160+34.73} \\ \\ Z=194.73\tan (15) \\ Z=52.18 \end{gathered}$

and the area of the triangle is

$\begin{gathered} A_{T2}=\frac{b\times h}{2} \\ \\ A_{T2}=\frac{Z\times(160+34.73)}{2}=\frac{52.18\times194.73}{2} \\ \\ A_{T2}=5080.5 \end{gathered}$

Total area

$\begin{gathered} A=A_{T1}+A_R+A_{T2} \\ A=3420.9+31520+5080.5 \\ A=40021.4 \end{gathered}$

the total area is 40,021.4 square feet