The first side of a triangle is twice the second, and the third is 20 feet less than three times the second. the perimeter is 106 feet. find the three sides.

Let's use 'x' for the first side, 'y' for the second, and 'z' for the third. Translate the statements into equations:

" The first side of a triangle is twice the second"

x = 2y

" the third is 20 feet less than three times the second"

z = 3y - 20

" the perimeter is 106 feet"

x + y + z = 106

Now let's plug in what 'x' and 'z' equal in the first two equations into the last equation:

2y + y + 3y - 20 = 106

Combine like terms (2y + y + 3y = 6y):

6y - 20 = 106

Add 20 to both sides:

6y = 126

Divide 6 to both sides:

y = 21

So the length of the second side is 21 feet. We can plug this into the first two equations to find the lengths of the other sides:

x = 2y

x = 2 (21)

Multiply:

x = 42

So the length of the first side is 42 feet.

z = 3y - 20

z = 3 (21) - 20

Multiply:

z = 63 - 20

Subtract:

z = 43

So the length of the third side is 43 feet.