The perimeter of a triangle is 189 feet. the longest side of the triangle is 36 feet shorter than twice the shortest side. the sum of the lengths of the two shorter sides is 49 feet more than the length of the longest side. find the lengths of the sides of the triangle.

Step 1: Let us assign variable for the unknown that we need to find in this problem:

Shortest side be ' x '

Second shortest side be ' y '

Longest side be ' z '

Note: As per the problem x < y < z

Step 2: Write equations based on the statement given

Statement 1 → "Perimeter of triangle is 189 feet"

Based on that above statement 1, we can set up an equation by adding all the sides and equating that to 189 feet.

x + y + z = 189 ⇒ 1st equation.

Statement 2 → "The longest side of the triangle is 36 feet shorter than twice the shortest side"

Based on the above statement 2, we can set up equation by multiplying 2 with x and taking away 36 from it and equating it to the longest side z.

z = 2x - 36 ⇒ 2nd equation

Statement 3 → "The sum of the lengths of the two shorter sides is 49 feet more than the length of the longest side"

Based on the above statement 3, we need to add x and y and equate it z added with 49.

x + y = z + 49 ⇒ 3rd equation.

Step 3: Now we need to solve the system of equation with three variables x, y and z using substitution method.

Substituting 3rd equation z + 49 in place of x + y in 1st equation we get

z + 49 + z = 189 {Combine like terms}

2z + 49 = 189 {Subtract 49 on both sides of the equation}

2z + 49 - 49 = 189 - 49

2z = 140 {Divide 2 on both sides}

z = 140/2 = 70 feet

z = 70 feet

Substitute 70 for z in the 2nd equation

z = 2x - 36

70 = 2x - 36 {Add 36 on both sides}

70 + 36 = 2x - 36 + 36 {Combine like terms in either side}

106 = 2x {Divide 2 on both sides}

106/2 = 2x/2 {Simplify fraction on either side of equation}

53 = x

Substitute 53 for x and 70 for z in 3rd equation

x + y = z + 49

53 + y = 70 + 49 {Combine like terms}

53 + y = 119 {Subtract 48 on both sides of the equation}

y = 119 - 53 = 66

Conclusion:

Shortest side of triangle = 53 feet

Second shortest of triangle = 66 feet

Longest side of the triangle = 70 feet