11. The Pentagon $P Q R S T$ is divided into four triangles with equal perimeters. The triangle $P Q R$ is equilateral. $P T U$, SUT and $R S U$ are congruent isosceles triangles. What is the ratio of the perimeter of the pentagon $P Q R S T$ to the perimeter of the triangle $P Q R$ ?
A $2: 1$
B $3: 2$
C $4: 3$
D $5: 3$
E $5: 2$
SOLUTION
D

Let $2 a$ be the side of equilateral triangle $P Q R$. Then $P Q R$ has a perimeter of $6 a$ and $P U$ has a length $a$.
Therefore, in order for the perimeter of triangle $P T U$ to be $6 a, P T$ must be $\frac{5}{2} a$.
Since the isosceles triangles are congruent $S T=P U$ and $R S=P T$. So the perimeter of the pentagon $P Q R S T$ is $2 a+2 a+\frac{5}{2} a+a+\frac{5}{2} a=10 a$. Therefore the ratio wanted is $10 a: 6 a$, that is $5: 3$.