Geometry

Solving Triangles

Sine Rule

         

In triangle ABCABC above, BAC=x=30,sinABC=14,BC=a=9,\angle BAC =x= 30^{\circ}, \sin \angle ABC = \frac{1}{4}, \lvert \overline{BC} \rvert =a= 9, where BC\lvert \overline{BC} \rvert is the length of BC.\overline{BC}. What is the value of AC2?\lvert \overline{AC} \rvert^2?

Triangle ABCABC has angles A=60\angle A = 60^{\circ} and B=45 \angle B = 45^{\circ} , and a side length a=216a = 21 \sqrt{6}. What is the side length bb?

Details and assumptions

aa and bb are the lengths of the sides opposite to the vertices AA and BB, respectively.

In triangle ABCABC above, we are given the side lengths as follows:BC=a,CA=b,AB=c.\lvert\overline{BC}\rvert=a, \lvert\overline{CA}\rvert=b, \lvert\overline{AB}\rvert=c. If ab:bc:ca=2:11:5,ab : bc : ca = 2 : 11 : 5, what is the value of sinA+sinBsinC?\frac{\sin A + \sin B}{\sin C}?

In triangle ABCABC above, ABC=x=30\angle ABC =x= 30^{\circ} and BC=a=6,CA=b=10.\lvert \overline{BC} \rvert =a= 6, \lvert \overline{CA} \rvert =b= 10. What is the value of (sinBAC)2?\left(\sin \angle BAC\right)^2?

In triangle ABCABC above, we are given the side lengths as follows:BC=a,CA=b,AB=c.\lvert\overline{BC}\rvert=a, \lvert\overline{CA}\rvert=b, \lvert\overline{AB}\rvert=c. If sinA:sinB=5:4\sin A : \sin B = 5 : 4 and  a:c=2:5,\ a : c = 2 : 5, what is a:b:c?a : b : c?

Note: The above diagram is not drawn to scale.

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