Problem 20 - Basic Technical Mathematics with Calculus

Question

Find the perimeter of each triangle. Isosceles triangle, equal sides of 2.45 in., third side of 3.22 in.

Step-by-Step Solution

Answer

The perimeter of the triangle is 8.12 inches.

Step 1: Identify the Sides

In an isosceles triangle, two sides are of equal length. Here, these two equal sides are each 2.45 inches long, and the third side is 3.22 inches.

Step 2: Perimeter Formula

The perimeter of a triangle is the sum of the lengths of all its sides. For this triangle, the formula is: \[ P = a + b + c\]where \(a\), \(b\), and \(c\) are the sides of the triangle.

Step 3: Substitute Values

Substitute the given side lengths into the perimeter formula. Let \(a = 2.45\) inches, \(b = 2.45\) inches, and \(c = 3.22\) inches: \[ P = 2.45 + 2.45 + 3.22\]

Step 4: Perform the Calculation

Add the values together:\[ P = 2.45 + 2.45 + 3.22 = 8.12\] inches.

Key Concepts

Isosceles Triangle

An isosceles triangle is a unique type of triangle where two of its three sides are of equal length. This characteristic gives the triangle some interesting properties. In our specific case, we deal with an isosceles triangle where the two equal sides are each 2.45 inches long. The key feature that makes this type of triangle notable is its symmetry. This symmetry also extends to the angles opposite the equal sides, which are also equal. Understanding these properties helps in identifying and solving problems related to isosceles triangles effectively.

Perimeter

The perimeter of any shape is the total distance around it. For triangles, this means adding up the lengths of all the sides. Calculation of the perimeter gives a sense of the size of the triangle's boundary. In the case of our isosceles triangle, the perimeter is found by adding the lengths of both equal sides with the third side. Understanding how to compute perimeter is very useful, not just for triangles but for many geometric shapes you encounter.

Triangle Sides

Triangles consist of three sides. In mathematical problems, particularly with isosceles triangles, identifying which sides are equal is crucial. In our example, the two equal sides are each 2.45 inches. The other side, known as the base, is 3.22 inches long. When you hear about triangle sides, it's important to picture them as the boundaries that make up the shape. Not only do they define its shape, but they are also key to calculations such as perimeter and area.

Mathematical Formula

Mathematical formulas are essential tools for solving geometry problems. For triangles, the perimeter formula provides a systematic way to add up all side lengths. Here, the formula to determine the perimeter of the triangle is:

• First, identify the three sides: two sides are 2.45 inches each, and the last is 3.22 inches.
• Then apply the formula: \[ P = a + b + c \] where \( a \), \( b \), and \( c \) are the side lengths.
• Finally, substitute the values: \[ P = 2.45 + 2.45 + 3.22 \]
• Calculate to find the perimeter: 8.12 inches.

Using such formulas not only simplifies computation but also provides a clear method to arrive at a solution.