Question

two students began a proof of the law of sines.
which student correctly started the proof, and what should that student do next to complete the proof?

Explanation:

Step1: Verify sine definitions

In triangle ABC, height \( h \) from C to AB. For angle A: opposite side = \( h \), hypotenuse = side opposite angle B (length \( a \)? No—correction: side opposite angle A is \( a \)? Wait, standard notation: side \( a = BC \), \( b = AC \), \( c = AB \). So \( \sin(A) = \frac{h}{BC} = \frac{h}{a} \)? No, Student 1: \( \sin(A)=\frac{h}{b} \) (if \( b=AC \), adjacent? No—wait, right triangle: \( \sin(A) = \frac{\text{opposite}}{\text{hypotenuse}} = \frac{h}{BC} = \frac{h}{a} \)? No, Student 2: \( \sin(A)=\frac{h}{a} \), \( \sin(B)=\frac{h}{b} \). Wait no—Student 1: \( \sin(A)=\frac{h}{b} \) (correct if \( b=BC \)? No, standard: side \( a \) opposite angle A, \( b \) opposite angle B. So \( \sin(A) = \frac{h}{a} \)? No, Student 1's first step: \( \sin(A)=\frac{h}{b} \) → solve for \( h = b\sin(A) \); \( \sin(B)=\frac{h}{a} \) → \( h=a\sin(B) \). Equate: \( b\sin(A)=a\sin(B) \) → \( \frac{\sin(A)}{a}=\frac{\sin(B)}{b} \), which is Law of Sines. Student 2's first step: \( \sin(A)=\frac{h}{a} \) (wrong, should be \( \frac{h}{b} \) if \( b \) is hypotenuse for angle A). So Student 1 is correct.

Step2: Next step for Student1

Law of Sines needs \( \frac{\sin(A)}{a}=\frac{\sin(B)}{b}=\frac{\sin(C)}{c} \). So add height from A or B to relate angle C.

Answer:

Student 1 correctly started the proof. Next, draw a height from another vertex (e.g., A to BC) to relate \( \sin(C) \) and complete the ratio \( \frac{\sin(A)}{a} = \frac{\sin(B)}{b} = \frac{\sin(C)}{c} \).