[tex]PQ=38[/tex]
Explanation
the perimeter of a triangle is the sum of the 3 lengths, so
[tex]\text{Perimeter}=length_1+length_2+length_3[/tex]
then,
Step 1
definde the perimeter of triangel SQT
[tex]\begin{gathered} \text{Perimeter}=length_1+length_2+length_3 \\ Perimeter_{SQT}=SQ+QT+TS \\ \text{replace} \\ 135=(2x-4)+QT+TS\rightarrow equation(1) \\ \end{gathered}[/tex]
Step 2
definde the perimeter of triangle PQR
[tex]\begin{gathered} \text{Perimeter}=length_1+length_2+length_3 \\ Perimeter_{PQR}=PQ+QR+RP \\ \text{replace} \\ 171=(x-9)+(2x-4)+QR+RP \\ 171=3x-13+QR+RP \\ 171+13=3x+QR+RP \\ 184=3x+QR+RP\rightarrow Equation\text{ (2)} \end{gathered}[/tex]
Step 3
as the triangles are congruent, the ratio of 2 perimeters must be the same, so
[tex]\begin{gathered} \frac{2x-4}{2x-4+x-9}=\frac{135}{171} \\ 171(2x-4)=135(3x-13) \\ 342x-684=405x-1755 \\ 342x-405x=-1755+684 \\ -63x=-1071 \\ x=\frac{-1071}{-63} \\ x=17 \end{gathered}[/tex]
Step 4
finally, replace x in PQ to find the measure
[tex]\begin{gathered} PQ=QS+SP \\ PQ=2x-4+x-9 \\ PQ=3x-13 \\ \text{replace} \\ PQ=3(17)-13 \\ PQ=38 \end{gathered}[/tex]
I hope this helps you
