To solve this problem, we will use the Law of Sines. The Law of Sines states that in any triangle, the ratio of the sine of an angle to the length of its opposite side is the same for all three angles.
Given:
- [tex]\(\angle P = 27^\circ\)[/tex]
- [tex]\(\angle R = 135^\circ\)[/tex]
- The side opposite [tex]\(\angle R\)[/tex] has a length of [tex]\(r = 9.5\)[/tex]
We need to find the length of side [tex]\(p\)[/tex], which is opposite [tex]\(\angle P\)[/tex].
The Law of Sines is written as:
[tex]\[
\frac{\sin P}{p} = \frac{\sin R}{r}
\][/tex]
Using the given angles and side length, we can write:
[tex]\[
\frac{\sin(27^\circ)}{p} = \frac{\sin(135^\circ)}{9.5}
\][/tex]
To solve for [tex]\(p\)[/tex], we rearrange the equation:
[tex]\[
p = \frac{9.5 \cdot \sin(27^\circ)}{\sin(135^\circ)}
\][/tex]
Given the numerical values:
[tex]\[
\sin(27^\circ) \approx 0.45399049973954675
\][/tex]
[tex]\[
\sin(135^\circ) \approx 0.7071067811865476
\][/tex]
We substitute these values into the equation:
[tex]\[
p = \frac{9.5 \cdot 0.45399049973954675}{0.7071067811865476}
\][/tex]
Performing the division:
[tex]\[
p \approx 6.0993754582419575
\][/tex]
Therefore, the length of side [tex]\(p\)[/tex] is approximately [tex]\(6.0993754582419575\)[/tex] units.
