In a 30 ^(∘) -60 ^(∘)-90 ^(∘) triangle, the hypotenuse h is 2 times the length of the shorter leg s, and the longer leg l is sqrt(3) times the length of the shorter leg.
Now let's take a look at the given picture. We will name the missing vertices.
First of all, we can see that SR and TU are congruent as SRTU is a rectangle. Therefore, the value of y is also 10.
Since △ URQ is a 30^(∘)-60^(∘)-90^(∘) triangle, the length of the hypotenuse is 2 times the length of the shorter leg.
RQ= 2RU
12=2( x)
x=6
The value of x is 6. We can add this information to our picture.
Next, let's recall that in a 30^(∘)-60^(∘)-90^(∘) triangle the length of the longer leg is sqrt(3) times the length of the shorter leg.
UQ=RU* sqrt(3)
z= 6sqrt(3)
The value of z is 6sqrt(3).
Let's notice that ST and RU are congruent. Therefore, the length of ST is 6. As we can see, △ PST is a 45^(∘)-45^(∘)-90^(∘) triangle, so the length of PT is also 6 and the length of PS is 6sqrt(2).
Finally, as we found all side lengths of trapezoid PQRS, we can evaluate its perimeter, P.
P=PS+SR+RQ+PQ
P=6sqrt(2)+10+12+(6+10+6sqrt(3))
P=38+6sqrt(2)+6sqrt(3)
The perimeter of trapezoid PQRS is 38+6sqrt(2)+6sqrt(3).
