Problem 27 - Fundamentals of Mathematics

Question

A person makes a mistake when copying information regarding a particular triangle. The copied information is as follows: Two sides of a triangle are the same length. The third side is 10 feet less than three times the length of one of the other sides. The perimeter of the triangle is 5 feet. What is the mistake?

Step-by-Step Solution

Answer

The mistake is that the third side length cannot be negative, indicating an error in the given relationships or perimeter.

Step 1: Define the Variables

Let's denote the length of the two equal sides of the triangle as \( x \) feet. The third side, according to the problem, would be \( 3x - 10 \) feet.

Step 2: Set Up the Perimeter Equation

The perimeter of a triangle is the sum of the lengths of all its sides. Therefore, the equation for the perimeter is \( x + x + (3x - 10) \). The problem states that the perimeter should be 5 feet, so the equation becomes \( x + x + (3x - 10) = 5 \).

Step 3: Simplify the Equation

Simplify the equation: \( 2x + (3x - 10) = 5 \). This simplifies to \( 5x - 10 = 5 \).

Step 4: Solve for x

To solve for \( x \), add 10 to both sides of the equation: \( 5x - 10 + 10 = 5 + 10 \), which simplifies to \( 5x = 15 \). Divide both sides by 5 to get \( x = 3 \).

Step 5: Calculate the Lengths of the Sides

With \( x = 3 \), the two equal sides are each \( 3 \) feet long. The third side is \( 3 \times 3 - 10 = 9 - 10 = -1 \) feet long, which is impossible.

Step 6: Identify the Mistake

The error lies in the fact that a side length of \(-1\) feet is not possible. The perimeter or the relationship of the third side's length is incorrect. Side lengths must be non-negative.

Key Concepts

Equilateral Triangle

An equilateral triangle is a special type of triangle where all three sides are of equal length. This means that in an equilateral triangle, each angle is also equal, specifically measuring 60 degrees.

• Equilateral triangles are both regular polygons and isosceles, as they have two or more equal sides.
• This property ensures that equilateral triangles are always symmetrical.

In this problem, however, the triangle is not equilateral since only two sides are mentioned to be equal. Understanding the unique properties of equilateral triangles helps in identifying non-equilateral types in various problems.

Perimeter Equation

The perimeter of a triangle is the total distance around the triangle. This is calculated by adding together the lengths of all three sides. Mathematically, it's represented as:
\[P = a + b + c\]where \(a\), \(b\), and \(c\) represent the side lengths.

• A correct perimeter equation accounts for all side lengths accurately.
• It provides a way to relate the side lengths to a known perimeter value.

In the problem, the equation set up identifies possible mistakes by showing impossible side lengths, emphasizing the importance of accurate calculations.

Side Length

Side length in triangles is crucial because it determines the type and possible dimensions of the triangle. Each side length must be a non-negative value since a physical measurement can't be negative. Here are important concepts to grasp:

• All sides of a triangle must adhere to the triangle inequality principle, which states that the sum of any two sides must be greater than the third side.
• If a calculated side comes out negative, it indicates an error or impossibility in the triangle's measurements.

In our exercise, the logic error arises when calculating a negative side length of \(-1\) feet, highlighting possible mistakes in the problem setup or given conditions.

Algebraic Equation

An algebraic equation uses mathematical symbols and variables to represent relationships between quantities. In problems involving triangles, such equations help in calculating unknown side lengths by setting up known values and relationships.
For instance, the provided problem sets up the equation:
\[x + x + (3x - 10) = 5\]

• This takes into account two equal sides and a third described by some functional relationship.
• Solving these equations requires simplifying: collecting like terms and isolating the variable on one side.

Correct usage of algebraic equations ensures logical consistency and accuracy in solving geometry problems. In this context, it exposed the error when the equation yielded a side length that was impossible.