Solving the Isosceles Triangle Puzzle: A Comprehensive Guide

Introduction

The isosceles triangle is a fascinating geometric shape characterized by its two equal sides and one base. This article aims to solve a specific problem related to an isosceles triangle, where the base is given and the perimeter is provided. The challenge lies in determining the length of the equal sides. We'll walk through the solution step-by-step, explaining each calculation and reasoning behind it.

The Problem Statement

A student was faced with a challenging problem: "The base of an isosceles triangle is 5 cm. The perimeter of the triangle is 13 4/10 cm. What is the length of the remaining equal sides?" However, the exact number to be subtracted from the perimeter was missing, making the problem incomplete. Despite this oversight, we can still provide a general solution based on the given information.

Step-by-Step Solution

Let's restate the problem and solve it methodically. We are given the base of the isosceles triangle, which is 5 cm, and the perimeter, which is 13 4/10 cm or 13.4 cm.

Step 1: Identify the Total of Both Equal Sides

The perimeter of a triangle is the sum of all its sides. For an isosceles triangle, this can be expressed as:

Perimeter Base 2 × Equal Side

Given:

Perimeter 13.4 cm

Base 5 cm

Replacing these values in the equation, we get:

13.4 5 2s, where 's' is the length of each of the sides.

Step 2: Isolate the Variable 's'

To find 's', we need to isolate the variable:

13.4 - 5 2s

8.4 2s

s 8.4 / 2

s 4.2 cm

This means each of the equal sides of the isosceles triangle measures 4.2 cm.

Alternative Methods and Solutions

Method 1: Simplified Expression

Some sources propose an alternative but equivalent expression:

1/2(13.4 - 5) 4.2 cm

This expression simplifies the calculation step without needing to swap terms or variables.

Method 2: Direct Perimeter Subtraction

Another common method involves directly subtracting the base from the perimeter and then dividing by 2:

(13.4 - 5) / 2 4.2 cm

This method simplifies the subtraction and division steps, making the calculation straightforward.

Common Mistakes to Avoid

It is important to avoid common mistakes such as:

Misreading the Problem: Confusing the given values or misinterpreting the problem statement. Incorrect Algebraic Manipulation: Failing to properly isolate variables or perform arithmetic operations correctly. Ignoring Units: Mixing units or misinterpreting the units provided.

Conclusion

Solving the isosceles triangle problem is straightforward when you follow the correct steps and methods. The key is to understand the relationship between the perimeter, base, and the equal sides of the triangle. Whether you use the equation, alternative methods, or direct subtraction, the goal remains the same: to find the length of the equal sides.

Additional Resources

For more detailed understanding and practice, consider exploring the following resources:

MathPlanet - Isosceles Triangle Khan Academy - Isosceles Triangle Properties Math Open Reference - Isosceles Triangle

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