In , let the sides opposite be . Determine the logical relationship between the statements: (Condition) (Conclusion) A. Sufficient but not necessary B. Necessary but not sufficient C. Necessary and sufficient D. Neither sufficient nor necessary

Triangle Solving Solution – Problem 23: Determine the logical relationship between the statements: (Condition) (Conclusion) A

Question

In $\triangle ABC$ , let the sides opposite $A,B,C$ be $a,b,c$ . Determine the logical relationship between the statements:

(Condition) $a\cos B=b\cos A$ (Conclusion) $A=B$

A. Sufficient but not necessary B. Necessary but not sufficient C. Necessary and sufficient D. Neither sufficient nor necessary

Step-by-step solution

Step 1. If $a\cos B=b\cos A$ , then using $\frac{a}{b}=\frac{\sin A}{\sin B}$ from the Law of Sines, we obtain $\sin A\cos B=\sin B\cos A$ .

Step 2. Assuming $A,B\ne\frac{\pi}{2}$ , divide both sides by $\cos A\cos B$ to get $\tan A=\tan B$ . Since $A$ and $B$ are interior angles of a triangle, $A=B$ . Thus the condition is sufficient.

Step 3. Conversely, if $A=B$ , then by the Law of Sines $a=b$ , and hence $a\cos B=b\cos A$ holds. Thus the condition is necessary.

Step 4. Therefore the condition is necessary and sufficient, so the correct choice is C.

Final answer

Marking scheme

1. Checkpoints (max 7 pts total)

Chain A: Law of Sines 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 [1 pt]: Considers branches and rejects invalid cases
Conclusion and verification [1 pt]: States the conclusion and checks against constraints
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 23: Determine the logical relationship between the statements: (Condition) (Conclusion) A