The distance AB across the river is approximately 171.4 meters
The known parameters are;
The distance BC laid off on one side of the river = 415 m
The measure of ∠B = 112.2°
The measure of angle ∠C = 18.3°
The unknown parameter;
The distance AB across the river
Strategy;
Taking the points A, B, and C, being the vertices of the triangle, ΔABC, and apply sine rule to find distance AB;
By the angle sum property, the measure of angle, ∠A = 180° - (∠B + ∠C)
∴ ∠A = 180° - (112.2° + 18.3°) = 49.5°
By sine rule, we get;
[tex]\mathbf{\dfrac{a}{sin (\alpha)} = \dfrac{b}{sin (\beta)} = \dfrac{c}{sin (\gamma)}}[/tex]
Therefore;
[tex]\mathbf {a = sin (\alpha) \times \dfrac{b}{sin (\beta)}}[/tex]
Plugging in α = AB, [tex]\alpha[/tex] = ∠C = 18.3°, b = BC = 415, β = ∠A = 49.5°, we get;
[tex]AB = sin (18.3 ^{\circ}) \times \dfrac{415}{sin (49.5^{\circ})} \approx 171.4[/tex]
The distance across the river, AB ≈ 171.4 m
Learn more about sine rule here;
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