Question

1. a tract of land has the shape of a trapezoid, as shown in figure. the lengths of three sides and the sizes of the two interior right angles are given. determine the two unknown interior angles and the length of the fourth side.

Explanation:

Step1: Find the horizontal difference

The two parallel sides (bases) of the trapezoid have lengths 120 m and 180 m. The horizontal difference between them is \( 180 - 120 = 60 \) m. The vertical side (height) is 100 m.

Step2: Find the length of the non - parallel side (hypotenuse of the right triangle)

We can use the Pythagorean theorem. Let the length of the unknown side be \( c \), the horizontal leg \( a = 60 \) m and the vertical leg \( b = 100 \) m. Then \( c=\sqrt{60^{2}+100^{2}}=\sqrt{3600 + 10000}=\sqrt{13600}\approx116.62 \) m.

Step3: Find the unknown angles

Let \( \theta \) be one of the unknown angles. We know that \( \tan\theta=\frac{100}{60}=\frac{5}{3}\approx1.6667 \), so \( \theta=\arctan(\frac{5}{3})\approx59.04^{\circ} \). The other unknown angle is \( 180^{\circ}- 59.04^{\circ}=120.96^{\circ} \) (since consecutive angles between the bases of a trapezoid are supplementary).

Answer:

The length of the fourth side is approximately \( 116.62 \) m, one unknown angle is approximately \( 59.04^{\circ} \) and the other is approximately \( 120.96^{\circ} \)