Geometry

Solving Triangles

Sine Rule

         

In triangle ABCABC above, ∠BAC=x=30∘,sin⁡∠ABC=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 ∣AC‾∣2?\lvert \overline{AC} \rvert^2?

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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.

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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 sin⁡A+sin⁡Bsin⁡C?\frac{\sin A + \sin B}{\sin C}?

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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 (sin⁡∠BAC)2?\left(\sin \angle BAC\right)^2?

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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 sin⁡A:sin⁡B=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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