Problem 36 - College Algebra with Modeling and Visualization

Question

Geometry The perimeter of a triangle is 105 inches. The longest side is 22 inches longer than the shortest side. The sum of the lengths of the two shorter sides is 15 inches more than the length of the longest side. Find the lengths of the sides of the triangle.

Step-by-Step Solution

Answer

The sides are 27, 31, and 47 inches.

Step 1: Understanding the Problem

We are given that the perimeter of the triangle is 105 inches. The longest side is 22 inches longer than the shortest side. Moreover, the sum of the lengths of the two shorter sides is 15 inches more than the longest side. We need to find the length of each side.

Step 2: Setting Up Variables

Let's denote the shortest side as \( x \). According to the problem, the longest side will be \( x + 22 \). Let's denote the middle side as \( y \). Then the perimeter equation is \( x + y + (x + 22) = 105 \).

Key Concepts

triangle perimeter

In geometry, the perimeter of a shape is the total length around its boundary. For a triangle, which has three sides, the perimeter is the sum of the lengths of all its sides. Knowing how to calculate the perimeter is crucial, as it helps in solving various geometry problems.

When you have the perimeter of a triangle, like 105 inches in this exercise, and additional information about the sides, you can use these details to find unknown side lengths. The basic formula for the perimeter of a triangle is:

• Perimeter = Side 1 + Side 2 + Side 3

By setting a variable for the shortest side, and using the given relationships, such as one side being a certain length longer than another, you can substitute these into the perimeter formula. This enables solving for the unknown values effectively.

geometry problems

Solving geometry problems often involves understanding and interpreting the relationships between different elements of a shape. In the case of triangles, being able to translate word problems into mathematical expressions is essential. This not only involves recognizing the data given but also identifying what additional information can be derived from it.

For instance, in this exercise, we understand that the longest side is 22 inches longer than the shortest. Therefore, if you choose to represent the shortest side as \( x \), the longest side can be expressed as \( x + 22 \).

Additionally, the sum of the lengths of the two shorter sides is 15 inches more than the longest side. This gives another equation where \( x + y = x + 22 + 15 \). Solving these equations requires careful attention, ensuring each piece of information is used accurately to form equations that reflect the problem's requirements. Understanding such relationships forms the basis of solving geometry problems.

algebra in geometry

Algebra can be a powerful tool when solving geometry problems. Using algebraic expressions to represent the unknown sides of a triangle can simplify complex problems. Assigning variables to different parts of the problem helps to express relationships and constraints algebraically.

In this particular problem, the variables help to set up equations based on the perimeter and the relationships between the sides. We used:

• \( x \) for the shortest side
• \( x + 22 \) for the longest side
• \( y \) for the middle side

Using these variables, we form equations that can be solved step by step. Such equations might look like:

• \( x + y + (x + 22) = 105 \)
• \( x + y = x + 22 + 15 \)

These equations allow us to calculate the lengths of each side by solving them using familiar algebraic methods. This combination of algebra and geometry provides a clear pathway to finding solutions.