Finding a Specific Solution

Oct 13, 2025 by windailys.com 16 views

Introduction

In this article, we will delve into the world of mathematics and explore a problem involving an isosceles triangle. The problem states that the triangle has two sides of equal length, denoted as $a$ , and a base of length $b$ . The perimeter of the triangle is given as 15.7 centimeters, and we are provided with the equation $2a + b = 15.7$ to model this information. Our goal is to solve for the values of $a$ and $b$ using algebraic techniques.

Understanding the Problem

Before we begin solving the problem, let's take a moment to understand the given information. We are dealing with an isosceles triangle, which means that two of its sides are of equal length. In this case, the two equal sides are denoted as $a$ , and the base of the triangle is denoted as $b$ . The perimeter of the triangle is given as 15.7 centimeters, which is the sum of the lengths of all three sides.

The Equation $2a + b = 15.7$

The equation $2a + b = 15.7$ is a linear equation that models the relationship between the lengths of the sides of the triangle. The equation states that the sum of twice the length of one side ( $2a$ ) and the length of the base ( $b$ ) is equal to 15.7 centimeters. This equation is a fundamental concept in algebra and is used to solve for the values of $a$ and $b$ .

Solving for $a$ and $b$

To solve for the values of $a$ and $b$ , we can use algebraic techniques. We can start by isolating one of the variables, either $a$ or $b$ . Let's isolate $b$ by subtracting $2a$ from both sides of the equation:

b = 15.7 - 2a

Now that we have isolated $b$ , we can substitute this expression into the original equation to get:

2a + (15.7 - 2a) = 15.7

Simplifying the equation, we get:

15.7 - 2a + 2a = 15.7

The $2a$ terms cancel out, leaving us with:

15.7 = 15.7

This equation is true for all values of $a$ , which means that we have an infinite number of solutions. However, we can still find a specific solution by choosing a value for $a$ . Let's choose $a = 5$ as a specific value.

Finding a Specific Solution

Now that we have chosen a value for $a$ , we can substitute this value into the equation to find the corresponding value of $b$ . Substituting $a = 5$ into the equation $b = 15.7 - 2a$ , we get:

b = 15.7 - 2(5)

Simplifying the equation, we get:

b = 15.7 - 10

b = 5.7

Therefore, the values of $a$ and $b$ are $a = 5$ and $b = 5.7$ .

Conclusion

In this article, we explored a problem involving an isosceles triangle and used algebraic techniques to solve for the values of $a$ and $b$ . We started by understanding the given information and then used the equation $2a + b = 15.7$ to model the relationship between the lengths of the sides of the triangle. We then solved for the values of $a$ and $b$ by isolating one of the variables and substituting the expression into the original equation. Finally, we found a specific solution by choosing a value for $a$ and substituting it into the equation to find the corresponding value of $b$ .

Additional Examples and Applications

The problem we solved in this article is a classic example of a linear equation and can be used to model a variety of real-world scenarios. For example, in architecture, the lengths of the sides of a building can be used to model the perimeter of the building. In engineering, the lengths of the sides of a bridge can be used to model the structural integrity of the bridge.

Real-World Applications

The problem we solved in this article has a variety of real-world applications. For example:

Architecture: The lengths of the sides of a building can be used to model the perimeter of the building.
Engineering: The lengths of the sides of a bridge can be used to model the structural integrity of the bridge.
Physics: The lengths of the sides of a triangle can be used to model the motion of an object.
Computer Science: The lengths of the sides of a triangle can be used to model the geometry of a computer-aided design (CAD) program.

Conclusion

Q: What is an isosceles triangle?

A: An isosceles triangle is a triangle with two sides of equal length. In this problem, the two equal sides are denoted as $a$ , and the base of the triangle is denoted as $b$ .

Q: What is the perimeter of the triangle?

A: The perimeter of the triangle is given as 15.7 centimeters, which is the sum of the lengths of all three sides.

Q: What is the equation $2a + b = 15.7$ ?

A: The equation $2a + b = 15.7$ is a linear equation that models the relationship between the lengths of the sides of the triangle. It states that the sum of twice the length of one side ( $2a$ ) and the length of the base ( $b$ ) is equal to 15.7 centimeters.

Q: How do we solve for the values of $a$ and $b$ ?

A: We can solve for the values of $a$ and $b$ by isolating one of the variables, either $a$ or $b$ . Let's isolate $b$ by subtracting $2a$ from both sides of the equation:

b = 15.7 - 2a

Q: What happens if we choose a value for $a$ ?

A: If we choose a value for $a$ , we can substitute this value into the equation to find the corresponding value of $b$ . For example, if we choose $a = 5$ , we can substitute this value into the equation $b = 15.7 - 2a$ to get:

b = 15.7 - 2(5)

Simplifying the equation, we get:

b = 15.7 - 10

b = 5.7

Therefore, the values of $a$ and $b$ are $a = 5$ and $b = 5.7$ .

Q: What are some real-world applications of the isosceles triangle problem?

A: The isosceles triangle problem has a variety of real-world applications, including:

Architecture: The lengths of the sides of a building can be used to model the perimeter of the building.
Engineering: The lengths of the sides of a bridge can be used to model the structural integrity of the bridge.
Physics: The lengths of the sides of a triangle can be used to model the motion of an object.
Computer Science: The lengths of the sides of a triangle can be used to model the geometry of a computer-aided design (CAD) program.

Q: What are some common mistakes to avoid when solving the isosceles triangle problem?

A: Some common mistakes to avoid when solving the isosceles triangle problem include:

Not isolating one of the variables: Make sure to isolate one of the variables, either $a$ or $b$ , by subtracting $2a$ from both sides of the equation.
Not choosing a value for $a$ : Make sure to choose a value for $a$ and substitute it into the equation to find the corresponding value of $b$ .
Not simplifying the equation: Make sure to simplify the equation by combining like terms.

Q: How can I practice solving the isosceles triangle problem?

A: You can practice solving the isosceles triangle problem by:

Working through multiple examples: Try solving the problem with different values of $a$ and $b$ .
Using online resources: There are many online resources available that provide practice problems and solutions for the isosceles triangle problem.
Seeking help from a teacher or tutor: If you are struggling to solve the problem, don't hesitate to seek help from a teacher or tutor.