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

In the following exercises, solve using the properties of triangles.
The perimeter of a triangle is 59 feet. One side of the triangle is 3 feet longer than the shortest
side. The third side is 5 feet longer than the shortest side. Find the length of each side.

Key Concepts

Relationships Between Triangle Sides

Understanding and utilizing the relationships between the sides of a triangle is key. When one side is stated to be a certain length longer than another, these relational descriptions can be translated into algebraic terms, enabling a systematic approach to finding the individual side lengths.

Solving Linear Equations

Once the side lengths are expressed algebraically and summed to equal the perimeter, a linear equation is formed. Solving this equation using standard algebraic techniques allows one to determine the value of the unknown variable, and subsequently, the lengths of all sides.

Triangle Perimeter

The perimeter of a triangle is defined as the sum of the lengths of all three sides. Knowing the total perimeter allows one to set up an equation that relates all side lengths, which is essential when some sides are described in relation to others.

Algebraic Representation of Triangle Sides

Using algebra, one can assign a variable to represent an unknown side length, and express the other sides in terms of this variable based on the given relationships. This conversion of geometric relations into algebraic expressions is a critical step in formulating the problem.

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