In the following exercises, solve using triangle properties....

In the following exercises, solve using triangle properties.
One side of a triangle is twice the shortest side. The third side is five feet more than the shortest side. The perimeter is 17 feet. Find the lengths of all three sides.

Key Concepts

Triangle Inequality Principle

The triangle inequality principle states that the sum of the lengths of any two sides of a triangle must be greater than the length of the remaining side. While not always explicitly used in forming the solution, it is an important property that any valid triangle must satisfy, ensuring that the calculated side lengths can indeed form a triangle.

Solving Linear Equations

Once the algebraic expressions are set up based on the given relationships and the perimeter, solving the resulting linear equation is essential. This involves combining like terms and isolating the variable to determine its value. This technique is crucial in translating the problem’s conditions into a solution.

Algebraic Representation of Geometric Relationships

This concept involves expressing the relationships among parts of a geometric figure using algebraic variables and equations. By assigning a variable to one of the sides, other sides can be expressed in relation to that variable. This transformation of geometric relationships into algebraic equations is fundamental in solving problems involving unknown side lengths.

Triangle Perimeter

The perimeter of a triangle is the sum of all its side lengths. In problems where the perimeter is given, setting up an equation that adds the expressions for all sides equal to the known perimeter is a key step. This provides a solvable equation that relates the sides together.

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