Determine 8133

Determine the distance between two places, M, and N, between which there is an obstacle so that place N is not visible from place M. The angles MAN = 130°, NBM = 109°, and the distances |AM| = 54, |BM| = 60, while the points A, B, and M lie on one straight line.

Correct answer:

x =  65.3113

Step-by-step explanation:

$$\begin{gathered} A=130{}^{\circ } \\ B=109{}^{\circ } \\ {A}_1=180-A=180-130=50{}^{\circ } \\ {B}_1=180-B=180-109=71{}^{\circ } \\ a=54 \\ b=60 \\ {x}_1=b-a+x_2 \\ {x}_2\tan (B_1)=x_1\tan (A_1) \\ {x}_2\tan (B_1)=(b-a+x_2)\tan (A_1) \\ {x}_2\tan (B_1)=(b-a)\cdot \tan (A_1)+x_2\tan (A_1) \\ {x}_2=\frac{(b-a)\cdot \tan A_1}{\tan B_1-\tan A_1}=\frac{(b-a)\cdot \tan 50°}{\tan 71°-\tan 50°}=\frac{(60-54)\cdot \tan 50°}{\tan 71°-\tan 50°}=\frac{(60-54)\cdot 1.191754}{2.904211-1.191754}=4.17559 \\ y=x_2\cdot \tan B_1=x_2\cdot \tan 71°=4.1756\cdot \tan 71°=4.1756\cdot 2.904211=12.1268 \\ x=\sqrt{y^2+(b+x_2{)}^2}=\sqrt{12.1268^2+(60+4.1756{)}^2}=65.3113 \end{gathered}$$

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