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!
[Math] How to calculate perimeter of Polygon with missing the length of one side
polygons
Related Solutions
Hint: Notice how the figure looks like a rectangle with an equilateral triangle cut out.
Let G and H, I be the midpoints of BC and DE, as O is the center OG, OH, OI are perpendicular to BC, DE, AF and let R be the radius of the circumscribed circle of the equilateral triangle,
You can find OG and OH by pyth. theorem,
$OG=\sqrt{R^2-18^2}$
$OH=\sqrt{R^2-18^2}$
$OI=\sqrt{R^2-18^2}$
by these
$OG=OH=OI=h$
Also by pyth. theorem you can get the lengths of BY, CZ, DZ, XE, XF, AY
$BY=18-\sqrt{12^2-h^2}$
$CZ=18-\sqrt{12^2-h^2}$
$DZ=18-\sqrt{12^2-h^2}$
$XE=18-\sqrt{12^2-h^2}$
$XF=18-\sqrt{12^2-h^2}$
$AY=18-\sqrt{12^2-h^2}$
With above you can find out $\triangle ABY, \triangle DCZ, \triangle XFE$ are equilateral and the side lengths of each other are equal.
By angle chasing you can find that the all angles of the hexagon is equal 120 and it is a regular hexagon.
As OB bisects $\angle ABC$, $\angle ABO=60$ and the one side of the hexagon will be equal to 12 (radius of the inscribed circle)
And the perimeter will be equal to = 12*6=72
Your answer is correct here is the proof for it
Best Answer
We can make use of the symmetry here and use Pythogoras theorm to Solve:
Observe that:
Similarly,
Now In right Traingle TOS, rt angled at O,
Hence, TS = 13
Now you can find the perimeter.