Solve each problem. See Example 1. The perimeter of a triangular plot of land is 2400 ft.The longest side is 200 ft less than twice the shortest. The middle side is 200 ft less than the longest side. Find the lengths of the three sides of the triangular plot.

Verified step by step guidance

1. Define variables for the sides of the triangle. Let the shortest side be \(x\) feet.
2. Express the longest side in terms of \(x\): the longest side is 200 feet less than twice the shortest, so it is \$2x - 200$ feet.
3. Express the middle side in terms of \(x\): the middle side is 200 feet less than the longest side, so it is \((2x - 200) - 200 = 2x - 400\) feet.
4. Write an equation for the perimeter by adding all three sides and setting the sum equal to 2400 feet: \(x + (2x - 200) + (2x - 400) = 2400\).
5. Simplify the equation and solve for \(x\). Once \(x\) is found, substitute back to find the lengths of the longest and middle sides.

Key Concepts

Setting Up Algebraic Expressions from Word Problems

This involves translating the given verbal information into algebraic expressions or equations. For example, identifying variables for unknown side lengths and expressing relationships like 'the longest side is 200 ft less than twice the shortest' as an equation helps form a solvable system.

Forming and Solving Systems of Linear Equations

Once expressions are established, you create equations based on the problem's conditions, such as the perimeter sum. Solving these simultaneous linear equations using substitution or elimination methods yields the values of the unknown variables.

Understanding Perimeter and Triangle Side Relationships

The perimeter is the sum of all side lengths of a triangle. Recognizing how the sides relate to each other and to the total perimeter is essential. This understanding ensures the equations correctly represent the problem's geometric constraints.