Equations, Inequalities, and Problem Solving

Solving Perimeter Problems Using Linear Equations

Perimeter problems often require translating a geometric scenario into an algebraic equation. In this example, the perimeter of a triangle is given, and the lengths of its sides are expressed in terms of a variable. The task is to find the value of the variable and then determine the length of each side.

• Perimeter Definition: The perimeter of a figure is the total distance around it, found by summing the lengths of all sides.

• Translating Words to Equations: Express the relationship in words, then translate to an equation. For a triangle:

  • In words: perimeter = side 1 + side 2 + side 3

  • In equation:

• Combining Like Terms: Remove parentheses and combine terms:

• Solving for the Variable: Use algebraic properties to isolate :

  • Add 10 to both sides:

  • Divide both sides by 18:

• Finding Side Lengths: Substitute back into the expressions for each side:

  • Side 1:

  • Side 2:

  • Side 3:

• Verification: Check that the sum matches the given perimeter:

Example: If a triangle has sides , , and and a perimeter of 224 feet, solve for and find the length of each side.

Step-by-step Solution:

• Write the equation:

• Combine like terms:

• Add 10:

• Divide by 18:

• Side 1: $13$ feet

• Side 2: $78$ feet

• Side 3: $133$ feet

Application: This method can be used for any polygon where side lengths are given in terms of a variable and the perimeter is known.