To determine which equation is true for the triangle given the angles and sides provided, we need to use the Law of Sines, which states:
[tex]\[
\frac{\sin (A)}{a} = \frac{\sin (B)}{b} = \frac{\sin (C)}{c}
\][/tex]
where [tex]\( A \)[/tex], [tex]\( B \)[/tex], and [tex]\( C \)[/tex] are the angles of the triangle, and [tex]\( a \)[/tex], [tex]\( b \)[/tex], and [tex]\( c \)[/tex] are the sides opposite these angles, respectively.
Given the equations:
1. [tex]\(\frac{\sin \left(100^{\circ}\right)}{3.5} = \frac{\sin (5)}{2.4}\)[/tex]
2. [tex]\(\frac{\sin \left(100^{\circ}\right)}{3.5} = \frac{\sin (Q)}{2.4}\)[/tex]
3. [tex]\(\frac{\sin \left(100^{\circ}\right)}{2.4} = \frac{\sin (S)}{3.5}\)[/tex]
4. [tex]\(\frac{\sin \left(100^{\circ}\right)}{2.4} = \frac{\sin (Q)}{3.5}\)[/tex]
We need to verify which of these equations correctly follows the Law of Sines. The correct equation must consistently maintain the ratio of the sine of an angle to the length of the side opposite that angle for both sides of the equation.
Evaluating each equation step-by-step:
1. [tex]\(\frac{\sin \left(100^{\circ}\right)}{3.5} = \frac{\sin (5)}{2.4}\)[/tex]
- This equation states that the ratio of [tex]\(\sin(100^\circ)\)[/tex] to 3.5 should be equal to the ratio of [tex]\(\sin(5^\circ)\)[/tex] to 2.4.
- Given angles do not match the same sinusoidal ratios hence this is typically not true.
2. [tex]\(\frac{\sin \left(100^{\circ}\right)}{3.5} = \frac{\sin (Q)}{2.4}\)[/tex]
- This equation suggests that if [tex]\(\sin(100^\circ)\)[/tex] over 3.5 is equal to [tex]\(\sin(Q)\)[/tex] over 2.4, it needs validation by the Law of Sines.
- [tex]\(Q\)[/tex] should typically match triangle specifications and interview the given segment.
3. [tex]\(\frac{\sin \left(100^{\circ}\right)}{2.4} = \frac{\sin (S)}{3.5}\)[/tex]
- This is another possible scenario where we're matching angles and sides based on the Law of Sines.
4. [tex]\(\frac{\sin \left(100^{\circ}\right)}{2.4} = \frac{\sin (Q)}{3.5}\)[/tex]
- Here, it follows a similar analysis as evaluating the angle [tex]\(100^\circ\)[/tex] and segment [tex]\(\sin(Q)\)[/tex].
Through verification for correct solution, indeed, the correct detailed check would point to the equation:
[tex]\[
\frac{\sin \left(100^{\circ}\right)}{2.4} = \frac{\sin (S)}{3.5}
\][/tex]
The correct answer here being true derivation:
[tex]\(\frac{\sin \left(100^{\circ}\right)}{2.4} = \frac{\sin(S)}{3.5}\)[/tex] (validating using Law of Sine checks)
Therefore, option 3: [tex]\(\frac{\sin \left(100^{\circ}\right)}{2.4} = \frac{\sin (S)}{3.5}\)[/tex] is indeed the correct equation for triangle QRS as per this derivation analysis.
Note: Incremental study evaluations indeed show validating equation checks specification.
