In the figure below, $\triangle DOG$, $\triangle ION$, and $\triangle IDO$ are congruent and isosceles, each with perimeter $55$. The quadrilaterals $FLIN,$ $DIES,$ and $DRAG$ are all squares. The perimeter of the $11$-sided polygon $DRAGONFLIES$ is $127$. What is the area of $DRAG$?

81 sq units.

Step-by-step explanation:

Given that Triangles DOG, ION and IDO are congruent and isosceles.

Let Sides NO = IO = DO = GO = units

Let the sides of squares FLIN, DIES and DRAG = units

So, NF = FL = LI = NI = IE = ES = SD = DI = DR = RA = AG = GD = units

Given that perimeter of each of Triangles DOG, ION and IDO = 55 units

Sum of sides of triangle =

and Perimeter of 11 sided polygon DRAGONFLIES = 127 units

The perimeter of the polygon includes the sides (only outer sides are included):

DR, RA, AG, GO, ON, NF, FL, LI, IE, ES and SD

Solving equations (2) and (1) by subtracting (1) from equation (2):

Area of a square DRAG = =