Answer:
[tex]P=25.97\ units[/tex]
Step-by-step explanation:
see the attached figure to better understand the problem
we know that
The perimeter of triangle PST is equal to
[tex]P=PT+PS+TS[/tex]
step 1
Find the length PS
In the right triangle PST
[tex]sin(48\°)=\frac{PT}{PS}[/tex] ---> opposite side angle of 48 degrees divide by the hypotenuse
substitute the given values
[tex]sin(48\°)=\frac{8}{PS}[/tex]
Solve for PS
[tex]PS=\frac{8}{sin(48\°)}[/tex]
[tex]PS=10.765\ units[/tex]
step 2
Find the length TS
In the right triangle PST
[tex]cos(48\°)=\frac{TS}{PS}[/tex] ---> adjacent side angle of 48 degrees divided by the hypotenuse
substitute the given values
[tex]cos(48\°)=\frac{TS}{10.77}[/tex]
Solve for TS
[tex]TS=cos(48\°)(10.765)[/tex]
[tex]TS=7.203\ units[/tex]
step 3
Find the perimeter
[tex]P=PT+PS+TS[/tex]
we have
[tex]PT=8\ units[/tex]
[tex]PS=10.765\ units[/tex]
[tex]TS=7.203\ units[/tex]
substitute
[tex]P=8+10.765+7.203=25.968\ units[/tex]
