How to calculate perimeter of Polygon with missing the length of one side?

Question

I have following sides(PQRST) of a Polygon where PQ=13, QR=22, RS=8, ST=?, PT= 10 ... i need to find out ST? i don't have any angle i just have the shape? And for calculating perimeter i need to find out the ST length of polygon!

This figure is showing all the sides that are given and side that i need to calculate for getting its perimeter.

Answer 1

Why don't you draw perpendicular from $T$ to $QR$ and from $S$ to $PQ$ ? Pythagoras Theorem.

Complete rectangles by drawing sides I told you about. Use the fact that opposite sides of rectangles are equal You should be able to get a right angled triangle with hypotenuse as $ST$ and other 2 sides as $13-8$ and $22-10$.

Answer 2

We can make use of the symmetry here and use Pythogoras theorm to Solve:

Observe that:

QR = PO = 22 {Opposite sides of a rectangle}
PT + TO = 22
TO = 22- PT = 22 - 10 = 12

Similarly,

PQ = OR = 13  {Opposite sides of a rectangle}
OS + SR = 13
OS = 13 - SR = 13 - 8 = 5

Now In right Traingle TOS, rt angled at O,

$TO^2 + OS^2 = ST^2  {Using Pythogoras Theorm}$
$12^2 + 5^2 = TS^2$
$TS^2 = 144 + 25 = 169$

Hence, TS = 13

Now you can find the perimeter.