To find the length of [tex]\( r \)[/tex] using the Law of Sines, we can use the formula:
[tex]\[
\frac{\sin \angle P}{p} = \frac{\sin \angle R}{r}
\][/tex]
Given:
- [tex]\(\angle P = 27^\circ\)[/tex]
- [tex]\(\angle R = 135^\circ\)[/tex]
- [tex]\( p = 9.5 \)[/tex]
We'll first need to express the sine values of the given angles.
To execute the steps:
1. Convert angles from degrees to radians:
- [tex]\(\angle P\)[/tex] in radians: [tex]\( \angle P_{\text{rad}} \approx 0.4712 \)[/tex]
- [tex]\(\angle R\)[/tex] in radians: [tex]\( \angle R_{\text{rad}} \approx 2.3562 \)[/tex]
2. Calculate the sine values:
- [tex]\( \sin(27^\circ) \approx 0.454 \)[/tex]
- [tex]\( \sin(135^\circ) \approx 0.707 \)[/tex]
3. Use the Law of Sines to create the equation:
[tex]\[
\frac{\sin(27^\circ)}{9.5} = \frac{\sin(135^\circ)}{r}
\][/tex]
4. Rearrange to solve for [tex]\( r \)[/tex]:
[tex]\[
r = \frac{9.5 \cdot \sin(135^\circ)}{\sin(27^\circ)}
\][/tex]
5. Substitute the sine values into the equation:
[tex]\[
r = \frac{9.5 \cdot 0.707}{0.454}
\][/tex]
6. Calculate [tex]\( r \)[/tex]:
[tex]\[
r \approx 14.7966
\][/tex]
Therefore, the length of [tex]\( r \)[/tex] is approximately [tex]\( 14.7966 \)[/tex].
