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Chapter 9: Problem 14

Solve. A triangle has sides of length 8 inches, 12 inches, and 10 inches. Find its perimeter.

Short Answer

Expert verified
The perimeter of the triangle is 30 inches.

Step by step solution

01

Identifying Focus

To solve for the perimeter of a triangle, you need to add up the lengths of all three sides of the triangle.
02

Setting Up the Perimeter Formula

The formula to find the perimeter of a triangle is the sum of the lengths of its sides. For any triangle with sides of lengths \(a\), \(b\), and \(c\), the perimeter \(P\) is given by \(P = a + b + c\).
03

Substituting Given Values

In this problem, the side lengths \(a = 8\) inches, \(b = 12\) inches, and \(c = 10\) inches. Substitute these values into the perimeter formula: \(P = 8 + 12 + 10\).
04

Calculating the Perimeter

Perform the addition: \(8 + 12 = 20\) and then \(20 + 10 = 30\). Thus, the perimeter of the triangle is 30 inches.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Understanding Triangle Sides
A triangle is a simple, three-sided polygon, and the length of its sides are crucial in determining many of its properties. Each side connects to two vertices, and the collective length of these sides gives us important dimensions like perimeter and area. In any given triangle, you can determine its type by looking at the relative lengths of its sides:
  • An equilateral triangle has all three sides of equal length.
  • An isosceles triangle has two sides that are equal in length.
  • A scalene triangle has all sides of different lengths.
In our example, the triangle has sides measuring 8 inches, 12 inches, and 10 inches, making it a scalene triangle. Understanding the different sides helps define the triangle's various properties, including how to calculate its perimeter.
Applying the Perimeter Formula
The perimeter of a triangle is simply the total distance around the triangle.
To find the perimeter, you add up the lengths of all three sides.
The formula used to calculate the perimeter of a triangle is given as:
\[ P = a + b + c \]where \(a\), \(b\), and \(c\) represent the lengths of the three sides.
Remember, the concept of a perimeter applies to all types of triangles, whether they are equilateral, isosceles, or scalene.
This formula is universal for triangles because:
  • It incorporates addition, which we'll look at in the next section.
  • It underscores the nature of a triangle's sides being connected in a sequence.
Importance of Addition in Calculating Perimeter
Addition is the fundamental arithmetic operation used in calculating a triangle's perimeter.
When identifying the triangle's perimeter, we apply addition to sum up the lengths of the triangle's sides.
Using the problem's values:
Initially, we add 8 inches and 12 inches, getting 20 inches.
Then, we add the third side鈥�10 inches鈥攖o our result of 20 inches, giving us a total of 30 inches.
Thus, the triangle's perimeter measures 30 inches.
Addition here connects each side's length to form a closed loop around the triangle.
  • It aggregates the segment lengths, crucial for calculating the boundary measure.
  • Simplifies the calculation by iterating through a basic arithmetic process.
Understanding this operation at its foundation helps prevent errors and supports a granular comprehension of geometric calculations.

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