Question

2
select the correct answer from the drop - down menu.
lisa used a surveyors measuring wheel to determine the side and diagonal lengths of an irregularly shaped plot of land, as shown.
image of a quadrilateral divided into two triangles, with side lengths 96 m, 126 m, 35 m, 111 m and diagonal 131 m
use the law of cosines to find the measures of interior angles in the figure, and then use the sine formula for the area of a triangle to approximate the total area of the plot of land.
the approximate area of the plot of land is drop - down menu with options 4887, 12377, 7483, 8340, 14885, 10886 square meters.
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Explanation:

Step1: Analyze the figure

The plot is divided into two triangles by the diagonal (131 m). Let's call the first triangle with sides 96 m, 126 m, 131 m and the second with sides 35 m, 111 m, 131 m.

Step2: Law of Cosines for angle in first triangle

For triangle with sides \(a = 96\), \(b = 126\), \(c = 131\), law of cosines: \(c^{2}=a^{2}+b^{2}-2ab\cos C\). So \(\cos C=\frac{a^{2}+b^{2}-c^{2}}{2ab}=\frac{96^{2}+126^{2}-131^{2}}{2\times96\times126}\). Calculate numerator: \(9216 + 15876-17161=7931\). Denominator: \(2\times96\times126 = 24192\). \(\cos C\approx\frac{7931}{24192}\approx0.328\), so \(C\approx70.8^{\circ}\). Area of first triangle: \(\frac{1}{2}ab\sin C=\frac{1}{2}\times96\times126\times\sin(70.8^{\circ})\). \(\sin(70.8^{\circ})\approx0.945\), so area \(\approx\frac{1}{2}\times96\times126\times0.945\approx96\times63\times0.945\approx96\times59.535\approx5715.36\).

Step3: Law of Cosines for angle in second triangle

For triangle with sides \(a = 35\), \(b = 111\), \(c = 131\), \(\cos C=\frac{35^{2}+111^{2}-131^{2}}{2\times35\times111}\). Numerator: \(1225 + 12321-17161=-3615\). Denominator: \(7770\). \(\cos C\approx - 0.465\), so \(C\approx117.7^{\circ}\). Area of second triangle: \(\frac{1}{2}\times35\times111\times\sin(117.7^{\circ})\). \(\sin(117.7^{\circ})=\sin(62.3^{\circ})\approx0.885\), so area \(\approx\frac{1}{2}\times35\times111\times0.885\approx35\times55.5\times0.885\approx35\times49.1175\approx1719.11\).

Step4: Total area

Total area \(\approx5715.36 + 1719.11\approx7434.47\), which is close to 7433 (option 7433). Wait, maybe more accurate calculation. Let's recalculate first triangle: \(96\times126 = 12096\), \(\frac{1}{2}\times12096=6048\). \(\sin(70.8^{\circ})\approx0.945\), so \(6048\times0.945\approx5715\). Second triangle: \(\frac{1}{2}\times35\times111 = 1942.5\). \(\sin(117.7^{\circ})\approx\sin(180 - 62.3)=\sin(62.3)\approx0.885\), so \(1942.5\times0.885\approx1719\). Total \(\approx5715 + 1719 = 7434\), which matches 7433 (due to rounding).

Answer:

7433