Question

a surveyor took some measurements of a piece of land. the owner needs to know the area of the land to determine the value. what is the area of the piece of land? (there is an image of a polygon with side lengths 21 ft, 12 ft, 30 ft, 22 ft and angles 90°, 54°, 54°, 90°)

Explanation:

Step1: Analyze the shape

The land can be divided into three triangles: a right triangle (21ft, 12ft, right angle), a triangle with sides 22ft, 22ft (isosceles with 54°), and a right triangle (30ft, 22ft, right angle? Wait, let's re - examine. Wait, the first triangle: legs 21ft and 12ft? Wait, no, the first triangle: base 21ft, height? Wait, the dashed line: first triangle: right triangle with legs 21ft and let's see, the other side? Wait, maybe better to split into three parts:

1. Triangle 1: right triangle with legs \(a = 21\) ft and \(b = 12\) ft? Wait, no, the angle is 90°, so area of triangle 1: \(A_1=\frac{1}{2}\times21\times12\)
2. Triangle 2: isosceles triangle with two sides 22ft and included angle 54°? Wait, no, the angle between the two 22ft sides? Wait, the angle is 54°, so area of triangle 2: \(A_2=\frac{1}{2}\times22\times22\times\sin(54^{\circ})\)
3. Triangle 3: right triangle with legs 30ft and 22ft? Wait, the angle is 90°, so area of triangle 3: \(A_3=\frac{1}{2}\times30\times22\)

Wait, maybe I mis - split. Let's look again. The figure:

- First triangle (top): right triangle, legs 21ft and 12ft? Wait, the vertical side 21ft, horizontal side 12ft? Wait, no, the dashed line. Wait, the first triangle: base 21ft, height 12ft? Wait, area of right triangle: \(A_1=\frac{1}{2}\times21\times12 = 126\) square feet.

- Second triangle: isosceles triangle with two sides 22ft and included angle 54°. The formula for area of a triangle with two sides \(a,b\) and included angle \(\theta\) is \(A=\frac{1}{2}ab\sin\theta\). So \(A_2=\frac{1}{2}\times22\times22\times\sin(54^{\circ})\). \(\sin(54^{\circ})\approx0.8090\), so \(A_2=\frac{1}{2}\times22\times22\times0.8090=\frac{1}{2}\times484\times0.8090 = 242\times0.8090\approx195.78\) square feet.

- Third triangle: right triangle, legs 30ft and 22ft? Wait, the vertical side 30ft, horizontal side 22ft? Wait, area of right triangle: \(A_3=\frac{1}{2}\times30\times22=330\) square feet.

Wait, no, maybe the second triangle is between the two dashed lines. Wait, another approach:

Wait, the figure can be divided into three triangles:

1. Triangle 1: right - angled at the corner, with legs \(l_1 = 21\) ft and \(l_2 = 12\) ft. Area \(A_1=\frac{1}{2}\times21\times12=126\) sq ft.

2. Triangle 2: two sides of length 22 ft and included angle \(54^{\circ}\). Area \(A_2=\frac{1}{2}\times22\times22\times\sin(54^{\circ})\). \(\sin(54^{\circ})\approx0.809\), so \(A_2=\frac{1}{2}\times484\times0.809\approx195.78\) sq ft.

3. Triangle 3: right - angled, with legs \(l_3 = 30\) ft and \(l_4 = 22\) ft. Area \(A_3=\frac{1}{2}\times30\times22 = 330\) sq ft.

Now, sum the areas: \(A = A_1+A_2+A_3\)

\(A=126 + 195.78+330=651.78\approx652\) square feet? Wait, maybe my splitting is wrong.

Wait, let's re - examine the figure. The three triangles:

1. Triangle 1: right triangle, legs 21 and 12. Area \(A_1=\frac{1}{2}\times21\times12 = 126\).

2. Triangle 2: isosceles triangle with sides 22, 22 and angle 54° between them. Area \(A_2=\frac{1}{2}\times22\times22\times\sin(54^{\circ})\approx\frac{1}{2}\times484\times0.809\approx195.8\).

3. Triangle 3: right triangle, legs 30 and 22. Area \(A_3=\frac{1}{2}\times30\times22 = 330\).

Total area \(A = 126+195.8 + 330=651.8\approx652\) square feet. Wait, but maybe the second triangle has sides 22 and 22, but the included angle is 54°, so that's correct.

Wait, another way: maybe the figure is composed of a right triangle (21,12), an isosceles triangle (22,22,54°), and a right triangle (30,22). Let's calculate each:

- \(A_1=\frac{1}{2}\times21\times12 = 126\)

- \(A_2=\frac{1}{2}\times22\times22\times\sin(54^{\circ})\). \(\sin(54^{\circ})\approx0.8090\), so \(A_2=\frac{1}{2}\times484\times0.8090 = 242\times0.8090\approx195.78\)

- \(A_3=\frac{1}{2}\times30\times22=330\)

Sum: \(126 + 195.78+330 = 651.78\approx652\) square feet.

Wait, but maybe I made a mistake in the second triangle. Wait, the two sides are 22 and 22, and the included angle is 54°, so the formula \(\frac{1}{2}ab\sin\theta\) is correct.

Alternatively, maybe the second triangle is not isosceles. Wait, the angle is 54°, and the two sides are 22 and 22? Wait, the length of the dashed line: if we have a triangle with two sides 22 and included angle 54°, then the area is \(\frac{1}{2}\times22\times22\times\sin(54^{\circ})\).

Another check: \(\sin(54^{\circ})\approx0.809\), so \(22\times22 = 484\), \(484\times0.809\approx391.56\), half of that is \(\approx195.78\), which matches.

The first triangle: \(\frac{1}{2}\times21\times12 = 126\), the third: \(\frac{1}{2}\times30\times22 = 330\). Sum: \(126 + 195.78+330=651.78\approx652\) square feet.

Step2: Recalculate if needed

Wait, maybe the second triangle has a different side. Wait, the length of the side between the two 54° angles: maybe the two sides are 22 and 22, and the included angle is 54°, so the area formula is correct.

Alternatively, maybe the figure is divided into three triangles:

1. Right triangle: legs 21 and 12: \(A_1=\frac{1}{2}\times21\times12 = 126\)

2. Triangle with sides 22, 22, and included angle 54°: \(A_2=\frac{1}{2}\times22\times22\times\sin(54^{\circ})\approx195.78\)

3. Right triangle: legs 30 and 22: \(A_3=\frac{1}{2}\times30\times22 = 330\)

Total area \(A = 126+195.78 + 330=651.78\approx652\) square feet.

Answer:

The area of the piece of land is approximately \(\boldsymbol{652}\) square feet (or more precisely \(651.78\) square feet).