Questions: The perimeter of a triangle is 71 cm. The longest side is 3 cm less than the sum of the other two sides. Twice the shortest side is 8 cm less than the longest side. Find the length of each side of the triangle. The shortest side is The middle side is The longest side is

Solution Steps

To solve this problem, we need to set up a system of equations based on the given conditions and solve for the lengths of the sides of the triangle.

1. Let the sides of the triangle be \(a\), \(b\), and \(c\) where \(a\) is the shortest side, \(b\) is the middle side, and \(c\) is the longest side.
2. The perimeter of the triangle is given as 71 cm, so \(a + b + c = 71\).
3. The longest side \(c\) is 3 cm less than the sum of the other two sides, so \(c = a + b - 3\).
4. Twice the shortest side is 8 cm less than the longest side, so \(2a = c - 8\).

We can use these equations to solve for \(a\), \(b\), and \(c\).

Let the sides of the triangle be \(a\), \(b\), and \(c\) where \(a\) is the shortest side, \(b\) is the middle side, and \(c\) is the longest side.

Given:

1. The perimeter of the triangle is 71 cm: \[ a + b + c = 71 \]
2. The longest side \(c\) is 3 cm less than the sum of the other two sides: \[ c = a + b - 3 \]
3. Twice the shortest side is 8 cm less than the longest side: \[ 2a = c - 8 \]

We solve the system of equations:

1. \(a + b + c = 71\)
2. \(c = a + b - 3\)
3. \(2a = c - 8\)

From equation (2): \[ c = a + b - 3 \]

Substitute \(c\) into equation (1): \[ a + b + (a + b - 3) = 71 \] \[ 2a + 2b - 3 = 71 \] \[ 2a + 2b = 74 \] \[ a + b = 37 \]

From equation (3): \[ 2a = (a + b - 3) - 8 \] \[ 2a = a + b - 11 \] \[ a = b - 11 \]

Substitute \(a = b - 11\) into \(a + b = 37\): \[ (b - 11) + b = 37 \] \[ 2b - 11 = 37 \] \[ 2b = 48 \] \[ b = 24 \]

Then: \[ a = b - 11 = 24 - 11 = 13 \]

Finally: \[ c = a + b - 3 = 13 + 24 - 3 = 34 \]

Final Answer