Question

14
select the correct answer from each drop - down menu.
a student began a proof of the law of sines using triangle xyz. his work is shown.

step 1: \\( \sin ( z ) = \frac { h } { x } \\)
\\( x \sin ( z ) = h \\)
the student has started the proof.
the next step in the proof is to
then the student should

Explanation:

Step1: Analyze the first - part

The student has correctly started the proof. In a right - triangle (formed by \(h\), \(x\), and the side adjacent to angle \(Z\)), by the definition of the sine function \(\sin\theta=\frac{\text{opposite}}{\text{hypotenuse}}\), for \(\theta = Z\), the opposite side is \(h\) and the hypotenuse is \(x\), so \(\sin(Z)=\frac{h}{x}\) and then \(x\sin(Z)=h\) is correct.

Step2: Find the next step

The next step in the proof is to write an equation for \(\sin(X)\). In the right - triangle formed by \(h\), \(z\), and the side adjacent to angle \(X\), using the sine function \(\sin(X)=\frac{h}{z}\), then \(z\sin(X)=h\).

Step3: Complete the proof

Since \(x\sin(Z)=h\) and \(z\sin(X)=h\), we can set them equal to each other. So then the student should set the two expressions for \(h\) equal to each other (\(x\sin(Z)=z\sin(X)\)) and then \(\frac{\sin(X)}{x}=\frac{\sin(Z)}{z}\) (by dividing both sides by \(xz\)).

Answer:

The student has correctly started the proof. The next step in the proof is to write an equation for \(\sin(X)\). Then the student should set the two expressions for \(h\) equal to each other.