Question

which equation is true for triangle qrs?

law of sines: \\(\frac{\sin(a)}{a} = \frac{\sin(b)}{b} = \frac{\sin(c)}{c}\\)

\\(\bigcirc\\) \\(\frac{\sin(100^{\circ})}{3.5} = \frac{\sin(s)}{2.4}\\)

\\(\bigcirc\\) \\(\frac{\sin(100^{\circ})}{3.5} = \frac{\sin(q)}{2.4}\\)

\\(\bigcirc\\) \\(\frac{\sin(100^{\circ})}{2.4} = \frac{\sin(s)}{3.5}\\)

\\(\bigcirc\\) \\(\frac{\sin(100^{\circ})}{2.4} = \frac{\sin(q)}{3.5}\\)

Explanation:

🆕 New Concept Discovered: Law of Sines
Relating angles to their opposite sides

Step 1: Identify opposite sides and angles

In any triangle, the Law of Sines relates the sine of an angle to the length of its opposite side:

$$ \frac{\sin(A)}{a} = \frac{\sin(B)}{b} = \frac{\sin(C)}{c} $$

Let's identify the angles and their opposite sides in triangle \(QRS\):

• The angle at vertex \(R\) is \(100^\circ\). Its opposite side is \(QS\), which has a length of \(3.5\).
• The angle at vertex \(S\) is \(S\). Its opposite side is \(QR\), which has a length of \(2.4\).
• The angle at vertex \(Q\) is \(Q\). Its opposite side is \(RS\), which is unlabeled.

Step 2: Set up the Law of Sines ratio

Using the known angle-side pairs, we can write the ratios:

$$ \frac{\sin(R)}{\text{opposite side}} = \frac{\sin(S)}{\text{opposite side}} $$

Substitute the known values into the formula:

$$ \frac{\sin(100^\circ)}{3.5} = \frac{\sin(S)}{2.4} $$

Step 3: Match with the given options

Let's compare our equation with the choices:

• Option 1: \(\frac{\sin(100^\circ)}{3.5} = \frac{\sin(S)}{2.4}\) (Matches our equation)
• Option 2: \(\frac{\sin(100^\circ)}{3.5} = \frac{\sin(Q)}{2.4}\) (Incorrect, \(2.4\) is opposite to angle \(S\), not \(Q\))
• Option 3: \(\frac{\sin(100^\circ)}{2.4} = \frac{\sin(S)}{3.5}\) (Incorrect, sides are paired with the wrong angles)
• Option 4: \(\frac{\sin(100^\circ)}{2.4} = \frac{\sin(Q)}{3.5}\) (Incorrect)

Answer:

The first option is correct:

$$ \frac{\sin(100^\circ)}{3.5} = \frac{\sin(S)}{2.4} $$