In , let the sides opposite be . Given , , and , find .

Triangle Solving Solution – Problem 29: find

Question

In $\triangle ABC$ , let the sides opposite $A,B,C$ be $a,b,c$ . Given $B=\frac{\pi}{3}$ , $b\sin C=\frac{3\sqrt{3}}{2}$ , and $a=2$ , find $b$ .

Step-by-step solution

Step 1. By the Law of Sines, $b\sin C=c\sin B$ . Since $b\sin C=\frac{3\sqrt{3}}{2}$ and $\sin B=\sin\frac{\pi}{3}=\frac{\sqrt{3}}{2}$ , we get $c=3$ .

Step 2. By the Law of Cosines, $b^{2}=a^{2}+c^{2}-2ac\cos B=4+9-2\cdot2\cdot3\cdot\frac12=7.$

Step 3. Hence $b=\sqrt{7}$ .

Final answer

$\sqrt{7}$

Marking scheme

1. Checkpoints (max 7 pts total)

Chain A: Law of Cosines approach

Set up side-angle relations [2 pts]: States and correctly advances the key derivation steps
Substitute and simplify [2 pts]: Substitutes correctly and simplifies accurately
Handle multiple cases / admissibility [2 pts]: Considers branches and rejects invalid cases
Final answer [1 pt]: Gives the correct final result (for multiple-choice, include the option letter)

2. Zero-credit items

Copies formulas without concrete substitution or derivation
Guesses the answer / provides only a conclusion with no reasoning
Uses an approach incompatible with the problem conditions, leading to an invalid conclusion

3. Deductions

Computation error [-1]: Incorrect algebraic/trigonometric manipulation
Logical gap [-1]: Missing a key equivalence step or a necessary condition check
Nonstandard final statement [-1]: Missing units/range/option letter or wrong answer format

Triangle Solving – Problem 29: find