Quadrilateral - irregular

Find the length of the side d = |AD| in quadrilateral ABCD: a= 35m, b= 120m, c=85m, angle ABC = 105 degrees, angle BCD= 72 degrees

Correct answer:

d = 113.0411 m

a = 35 m b = 120 m c = 85 m β = 105^{\circ} γ = 72^{\circ} B_{2} = 180 - β = 180 - 105 = 75^{\circ} C_{2} = B_{2} + (180 - γ) = 75 + (180 - 72) = 183^{\circ} A = (0, 0) B = (a, 0) C = (C_{x}, C_{y}) D = (D_{x}, D_{y}) C_{x} = B_{x} + b \cdot cos B_{2} = B_{x} + b \cdot cos 75 ° = 35 + 120 \cdot cos 75 ° = 35 + 120 \cdot 0.258819 = 66.05829 C_{y} = B_{y} + b \cdot sin B_{2} = B_{y} + b \cdot sin 75 ° = 0 + 120 \cdot sin 75 ° = 0 + 120 \cdot 0.965926 = 115.9111 D_{x} = C_{x} + c \cdot cos C_{2} = C_{x} + c \cdot cos 183 ° = 66.0583 + 85 \cdot cos 183 ° = 66.0583 + 85 \cdot (- 0.99863) = - 18.82523 D_{y} = C_{y} + c \cdot sin C_{2} = C_{y} + c \cdot sin 183 ° = 115.9111 + 85 \cdot sin 183 ° = 115.9111 + 85 \cdot (- 0.052336) = 111.46254 d = d i s t (A, D) = ∣ A D ∣ = (A_{x} - D_{x})^{2} + (A_{y} - D_{y})^{2} = (0 - (- 18.8252))^{2} + (0 - 111.4625)^{2} = 113.0411 m

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