Question

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?

Answer 1

81

Explanation:

dont ask, i just know

Answer 2

81 sq units.

Explanation:

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

Let Sides NO = IO = DO = GO =
`x` units

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

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

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

Sum of sides of triangle =
`2x+a=55 ...... (1)`

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

`2x+9a = 127......(2)`

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

`8a = 72\\\Rightarrow a = 9 units`

Area of a square DRAG =
`Side^2` =
`9^2 = \bold{81\ sq\ units}`