What total length of ribbon is needed to line the two long sides of the hat?

Total length of ribbon to line the long sides will be 22 in

Step-by-step explanation:

In the given figure of kite if we draw a line "h" vertically will form two right angle triangles.

Then by applying Pythagoras theorem in both the triangles.

h²= (3x + 5)² + (2x + 1)² (1)

h²= (5x + 1)² + 5² (2)

By equating both the equations

(3x + 5)² + (2x + 1)² = (5x + 1)² + 25

9x² + 25 + 30x + 4x² + 4x + 1 = 25x² + 10x + 1 + 25

13x² + 34x + 26 = 25x² + 10x + 26

13x² - 25x² + 34x - 10x + 26 - 26 = 0

- 12x² + 24x = 0

12x² - 24x = 0

x² - 2x = 0

x(x - 2) = 0

x = 2

Now we will put x = 2 in the measure of sides of the kite.

Side 1 = (3x + 5)

          = 3×2 + 5

          = 11

Side 2 = (5x + 1)

          = 5×2 + 1

          = 11

Side 3 = (2x + 1)

          = 2×2 + 1

          = 5

Therefore, Total length of ribbon to line the long sides will be = 11 + 11

= 22 in.