Common Variations of the Problem

Oct 13, 2025 by miodeya.com 68 views

Introduction

In the world of geometry, triangles are fundamental shapes that have been studied for centuries. With their three sides and three angles, triangles are a staple of mathematics, appearing in various fields such as physics, engineering, and architecture. In this article, we will delve into a specific problem involving a triangle, where the shortest side is given as $s$ centimeters long. The longest side is 8 cm longer than the shortest side, and the third side is 12 cm long. The perimeter of the triangle is 38 cm. Our goal is to find the length of the shortest side, which is represented by the variable $s$ .

Understanding the Problem

Let's break down the information given in the problem:

The shortest side of the triangle is $s$ centimeters long.
The longest side is 8 cm longer than the shortest side, which means it is $s + 8$ cm long.
The third side is 12 cm long.
The perimeter of the triangle is 38 cm.

The perimeter of a triangle is the sum of the lengths of its three sides. Therefore, we can write an equation to represent the perimeter:

$s + (s + 8) + 12 = 38$

Solving the Equation

Now that we have the equation, let's solve for $s$ . To do this, we will combine like terms and isolate the variable $s$ .

$s + s + 8 + 12 = 38$

Combine like terms:

$2s + 20 = 38$

Subtract 20 from both sides:

$2s = 18$

Divide both sides by 2:

$s = 9$

Conclusion

Therefore, the length of the shortest side of the triangle is 9 cm. This is the solution to the problem, and it represents the value of the variable $s$ .

Discussion

In this problem, we used basic algebraic techniques to solve for the length of the shortest side of the triangle. We started with an equation representing the perimeter of the triangle and then solved for the variable $s$ . This problem is a great example of how mathematics can be used to solve real-world problems, and it highlights the importance of algebra in geometry.

Real-World Applications

The problem we solved is a simple example of how mathematics can be used to solve real-world problems. In the real world, triangles are used in various applications, such as:

Building design: Triangles are used in the design of buildings to ensure that they are structurally sound.
Engineering: Triangles are used in engineering to calculate stresses and strains on materials.
Physics: Triangles are used in physics to calculate distances and velocities.

Final Thoughts

In conclusion, the problem we solved is a great example of how mathematics can be used to solve real-world problems. By using basic algebraic techniques, we were able to find the length of the shortest side of the triangle. This problem highlights the importance of algebra in geometry and demonstrates how mathematics can be used to solve real-world problems.

Additional Resources

For those who want to learn more about triangles and geometry, here are some additional resources:

Khan Academy: Geometry
Mathway: Triangle Problems
Wolfram Alpha: Triangle Calculator

References

"Geometry" by Michael Artin
"Algebra" by Michael Artin
"Mathematics for the Nonmathematician" by Morris Kline

Introduction

In our previous article, we explored the problem of finding the length of the shortest side of a triangle, given that the longest side is 8 cm longer than the shortest side, and the third side is 12 cm long. The perimeter of the triangle is 38 cm. We solved for the length of the shortest side, which is represented by the variable $s$ . In this article, we will answer some frequently asked questions related to this problem.

Q&A

Q: What is the perimeter of a triangle?

A: The perimeter of a triangle is the sum of the lengths of its three sides.

Q: How do I find the length of the shortest side of a triangle?

A: To find the length of the shortest side of a triangle, you can use the formula:

$s + (s + 8) + 12 = 38$

where $s$ is the length of the shortest side.

Q: What if the longest side is not 8 cm longer than the shortest side?

A: If the longest side is not 8 cm longer than the shortest side, you will need to use a different equation to solve for the length of the shortest side.

Q: Can I use this method to find the length of the longest side?

A: Yes, you can use this method to find the length of the longest side by substituting the value of the shortest side into the equation.

Q: What if the third side is not 12 cm long?

A: If the third side is not 12 cm long, you will need to use a different equation to solve for the length of the shortest side.

Q: Can I use this method to find the length of the third side?

A: Yes, you can use this method to find the length of the third side by substituting the value of the shortest side into the equation.

Q: What if the perimeter of the triangle is not 38 cm?

A: If the perimeter of the triangle is not 38 cm, you will need to use a different equation to solve for the length of the shortest side.

Q: Can I use this method to find the length of the shortest side of a right triangle?

A: Yes, you can use this method to find the length of the shortest side of a right triangle.

Q: What if the triangle is not a right triangle?

A: If the triangle is not a right triangle, you will need to use a different equation to solve for the length of the shortest side.

Conclusion

In this article, we answered some frequently asked questions related to the problem of finding the length of the shortest side of a triangle. We provided step-by-step instructions on how to solve for the length of the shortest side, and we discussed some common variations of the problem.

Additional Resources

For those who want to learn more about triangles and geometry, here are some additional resources:

Khan Academy: Geometry
Mathway: Triangle Problems
Wolfram Alpha: Triangle Calculator

References

"Geometry" by Michael Artin
"Algebra" by Michael Artin
"Mathematics for the Nonmathematician" by Morris Kline

Final Thoughts

In conclusion, the problem of finding the length of the shortest side of a triangle is a classic example of how mathematics can be used to solve real-world problems. By using basic algebraic techniques, we can find the length of the shortest side of a triangle, and we can use this method to solve a variety of problems involving triangles.

Common Variations of the Problem

Here are some common variations of the problem:

The longest side is 10 cm longer than the shortest side.
The third side is 15 cm long.
The perimeter of the triangle is 42 cm.
The triangle is a right triangle.

These variations can be solved using the same method as the original problem.

Solutions to Common Variations of the Problem

Here are the solutions to the common variations of the problem:

The longest side is 10 cm longer than the shortest side: $s + (s + 10) + 12 = 42$ $2s + 22 = 42$ $2s = 20$ $s = 10$
The third side is 15 cm long: $s + (s + 8) + 15 = 38$ $2s + 23 = 38$ $2s = 15$ $s = 7.5$
The perimeter of the triangle is 42 cm: $s + (s + 8) + 12 = 42$ $2s + 20 = 42$ $2s = 22$ $s = 11$
The triangle is a right triangle: $s + (s + 8) + 12 = 38$ $2s + 20 = 38$ $2s = 18$ $s = 9$

These solutions demonstrate how the method can be used to solve a variety of problems involving triangles.