Problem 24 - Prealgebra

Question

Two sides of a triangle are equal in length, and the third side is 10 inches. If the perimeter is 26 inches, how long are the two equal sides?

Step-by-Step Solution

Answer

The two equal sides of the triangle are each 8 inches long.

Step 1: Define the Variables

Let the length of the two equal sides of the triangle be denoted by \( x \). Since the third side is given as 10 inches, we can represent the sides of the triangle as \( x, x, 10 \).

Step 2: Write the Equation for the Perimeter

The perimeter of a triangle is the sum of the lengths of its sides. Therefore, the equation for the perimeter is: \( x + x + 10 = 26 \).

Step 3: Simplify the Equation

Combine like terms in the equation: \( 2x + 10 = 26 \).

Step 4: Solve for x

Subtract 10 from both sides to isolate terms with \( x \): \( 2x = 16 \). Then, divide both sides by 2 to solve for \( x \): \( x = 8 \).

Step 5: Verify the Solution

To ensure the solution is correct, substitute \( x = 8 \) back into the equation for the perimeter to check: \( 8 + 8 + 10 \) should equal the perimeter of 26. Indeed, \( 8 + 8 + 10 = 26 \), confirming our solution is correct.

Key Concepts

Isosceles Triangle

An isosceles triangle is a special type of triangle that has exactly two sides of equal length. These equal sides are known as the "legs" of the triangle, while the remaining side is called the "base." The angles opposite to the equal sides are also equal in measure.
In our exercise, identifying the triangle as isosceles is crucial. Knowing that two sides are equal helps us set a balanced equation to solve for the unknown length of these sides.
Understanding the properties of an isosceles triangle is important in geometry, as it forms the basis for solving many related problems. This type of triangle has characteristics that often simplify complex problems and allow for more straightforward algebraic solutions.

Algebraic Equation

An algebraic equation is a mathematical statement that uses variables, numbers, and operations to express relationships between different quantities. It often involves finding the unknown values that satisfy the given relationship.
In this problem, we utilized an algebraic equation to represent the perimeter relationship of the isosceles triangle. We set the equation as: \[ x + x + 10 = 26 \] where \( x \) represents the length of the two equal sides. Writing this equation is a crucial step since it translates the geometrical problem into an algebraic one. This approach makes it easier to manipulate the equation and find the unknown variable, \( x \).
Mastering the art of forming and solving algebraic equations is key in various math problems, not just in geometry.

Problem Solving

Problem solving in mathematics involves a systematic approach to finding solutions to questions or challenges. It requires both creative thinking and logical reasoning, making use of various strategies to reach a solution.
In this triangle problem, problem solving entailed several steps: defining variables, setting up an equation, and performing mathematical operations to isolate and solve for the unknown. Breaking down the problem into manageable steps and verifying the final answer is part of effective problem-solving techniques.

• Define variables: Represent unknown quantities using symbols or letters.
• Write equations: Use known information to create mathematical statements.
• Solve equations: Apply algebraic techniques to find the value of unknowns.
• Verify solutions: Ensure the solution satisfies the original conditions of the problem.

Geometry

Geometry is a branch of mathematics focused on the study of shapes, sizes, and properties of space. It involves understanding and solving problems related to points, lines, surfaces, and solids.
In this exercise, geometry helps us understand the properties of an isosceles triangle, which guided us in forming the problem's algebraic equation for the perimeter. The problem required an appreciation of geometric principles such as the definition of an isosceles triangle and the sum of side lengths to find the solution.
Geometry also underpins many real-world applications, from architecture to engineering, where spatial understanding is crucial. Recognizing the relevance of geometric principles develops logical thinking and problem-solving skills that are broadly applicable beyond mathematics.