Problem 19 - Basic Technical Mathematics with Calculus

Question

Find the perimeter of each triangle. An equilateral triangle of sides \(21.5 \mathrm{cm}\)

Step-by-Step Solution

Answer

The perimeter of the equilateral triangle is 64.5 cm.

Step 1: Understand the Problem

We are asked to find the perimeter of an equilateral triangle with each side measuring \(21.5\, \text{cm}\).

Step 2: Recall the Formula for Perimeter

The perimeter of an equilateral triangle (a triangle with all sides equal) is given by the formula \( P = 3a \), where \( a \) is the length of one side.

Step 3: Substitute the Known Values

Substitute \( a = 21.5 \) cm into the formula: \[ P = 3 imes 21.5 \]

Step 4: Calculate the Perimeter

Now calculate the expression:\[ P = 3 imes 21.5 = 64.5 \]Thus, the perimeter of the triangle is \( 64.5 \) cm.

Key Concepts

Equilateral Triangle

An equilateral triangle is a special type of triangle where all three sides are identical in length. This symmetry gives equilateral triangles unique properties. For instance, not only are the sides equal, but the angles are also the same, each measuring precisely 60 degrees. This characteristic makes it easier to calculate the perimeter, as you only need the length of one side. By understanding the uniformity of an equilateral triangle, solving problems such as finding its perimeter becomes straightforward. Knowing that each side is equal simplifies calculations and lets you apply the same principles across any equilateral triangle.

Geometry

Geometry is a branch of mathematics that studies the sizes, shapes, properties, and dimensions of different figures. In particular, geometry helps us understand how figures like triangles behave. Triangles are basic geometric shapes with three sides and three angles, forming the foundation for understanding more complex shapes. Within geometry, various special triangles like equilateral, isosceles, and scalene contribute to the diversity of problems and solutions. Learning geometry allows you to comprehend fundamental concepts that you can use in everyday life, from understanding maps to designing objects. It provides tools for measuring space, calculating distances, and solving practical problems.

Triangle Properties

Triangles are fascinating shapes with various properties based on their side lengths and angles. The basic properties include:

• Sum of Angles: The internal angles of any triangle add up to 180 degrees.
• Types of Triangles: Based on sides, there are equilateral, isosceles (two sides equal), and scalene (all sides different); based on angles, there are acute (all angles less than 90°), right (one angle is 90°), and obtuse (one angle more than 90°) triangles.
• Congruency: Two triangles are congruent if they have the same size and shape, allowing for problems to be solved using this one-to-one correspondence.

A solid understanding of these properties helps in recognizing and solving for unknowns such as side lengths, angles, and perimeters in varied problems. Knowing these truths about triangles makes it easier to work with them in both theoretical and practical contexts, enhancing your mathematical and spatial reasoning skills.