Which Classification Best Represents A Triangle With Side Lengths 6 Cm , 10 Cm , And 12 Cm 6 \, \text{cm}, 10 \, \text{cm}, \text{and } 12 \, \text{cm} 6 Cm , 10 Cm , And 12 Cm ?A. Acute, Because 6 2 + 10 2 \textless 12 2 6^2 + 10^2 \ \textless \ 12^2 6 2 + 1 0 2 \textless 1 2 2 B. Acute, Because $6 + 10 \ \textgreater \

Introduction

In geometry, triangles are classified based on the measure of their angles. The three main types of triangles are acute, right, and obtuse triangles. Understanding the properties of these triangles is essential in mathematics, particularly in trigonometry and geometry. In this article, we will explore the classification of triangles and determine which type best represents a triangle with side lengths $6 \, \text{cm}, 10 \, \text{cm}, \text{and } 12 \, \text{cm}$.

Understanding Acute, Right, and Obtuse Triangles

An acute triangle is a triangle with all angles measuring less than $90^\circ$. A right triangle is a triangle with one angle measuring exactly $90^\circ$. An obtuse triangle is a triangle with one angle measuring greater than $90^\circ$.

Properties of Acute Triangles

An acute triangle has the following properties:

• All angles measure less than $90^\circ$.
• The sum of the squares of the two shorter sides is greater than the square of the longest side.
• The triangle has no right angles.

Properties of Right Triangles

A right triangle has the following properties:

• One angle measures exactly $90^\circ$.
• The sum of the squares of the two shorter sides is equal to the square of the longest side.
• The triangle has one right angle.

Properties of Obtuse Triangles

An obtuse triangle has the following properties:

• One angle measures greater than $90^\circ$.
• The sum of the squares of the two shorter sides is less than the square of the longest side.
• The triangle has no acute angles.

Determining the Type of Triangle

To determine the type of triangle, we need to examine the side lengths and the relationships between them. Let's consider the given triangle with side lengths $6 \, \text{cm}, 10 \, \text{cm}, \text{and } 12 \, \text{cm}$.

Analyzing the Side Lengths

We can analyze the side lengths by comparing the squares of the two shorter sides to the square of the longest side. If the sum of the squares of the two shorter sides is greater than the square of the longest side, the triangle is acute. If the sum of the squares of the two shorter sides is equal to the square of the longest side, the triangle is right. If the sum of the squares of the two shorter sides is less than the square of the longest side, the triangle is obtuse.

Calculating the Squares of the Side Lengths

Let's calculate the squares of the side lengths:

• $6^2 = 36$
• $10^2 = 100$
• $12^2 = 144$

Comparing the Squares

Now, let's compare the sum of the squares of the two shorter sides to the square of the longest side:

• $6^2 + 10^2 = 36 + 100 = 136$
• $12^2 = 144$

Since $6^2 + 10^2 \ \textless \ 12^2$, the triangle is not acute. However, option A states that the triangle is acute because $6^2 + 10^2 \ \textless \ 12^2$. This is incorrect, as the triangle is not acute.

Conclusion

In conclusion, the triangle with side lengths $6 \, \text{cm}, 10 \, \text{cm}, \text{and } 12 \, \text{cm}$ is not acute, as the sum of the squares of the two shorter sides is less than the square of the longest side. However, option A states that the triangle is acute because $6^2 + 10^2 \ \textless \ 12^2$. This is incorrect, as the triangle is not acute.

Discussion

The discussion surrounding the classification of triangles is an essential aspect of mathematics. Understanding the properties of acute, right, and obtuse triangles is crucial in trigonometry and geometry. In this article, we explored the classification of triangles and determined which type best represents a triangle with side lengths $6 \, \text{cm}, 10 \, \text{cm}, \text{and } 12 \, \text{cm}$. We also analyzed the side lengths and the relationships between them to determine the type of triangle.

References

• "Geometry: A Comprehensive Introduction" by Dan Pedoe
• "Trigonometry: A Unit Circle Approach" by Michael Corral
• "Mathematics for the Nonmathematician" by Morris Kline

Further Reading

For further reading on the classification of triangles, we recommend the following resources:

• "Triangle Classification" by Math Open Reference
• "Types of Triangles" by Purplemath
• "Triangle Properties" by Math Is Fun

Conclusion

Q: What is the difference between an acute triangle and a right triangle?

A: An acute triangle is a triangle with all angles measuring less than $90^\circ$, while a right triangle is a triangle with one angle measuring exactly $90^\circ$.

Q: How do I determine if a triangle is acute or right?

A: To determine if a triangle is acute or right, you need to examine the side lengths and the relationships between them. If the sum of the squares of the two shorter sides is greater than the square of the longest side, the triangle is acute. If the sum of the squares of the two shorter sides is equal to the square of the longest side, the triangle is right.

Q: What is the property of an obtuse triangle?

A: An obtuse triangle has one angle measuring greater than $90^\circ$. The sum of the squares of the two shorter sides is less than the square of the longest side.

Q: How do I determine if a triangle is obtuse?

A: To determine if a triangle is obtuse, you need to examine the side lengths and the relationships between them. If the sum of the squares of the two shorter sides is less than the square of the longest side, the triangle is obtuse.

Q: What is the relationship between the side lengths of a triangle and its classification?

A: The relationship between the side lengths of a triangle and its classification is as follows:

• If the sum of the squares of the two shorter sides is greater than the square of the longest side, the triangle is acute.
• If the sum of the squares of the two shorter sides is equal to the square of the longest side, the triangle is right.
• If the sum of the squares of the two shorter sides is less than the square of the longest side, the triangle is obtuse.

Q: Can a triangle be both acute and right?

A: No, a triangle cannot be both acute and right. A triangle can be either acute or right, but not both.

Q: Can a triangle be both right and obtuse?

A: No, a triangle cannot be both right and obtuse. A triangle can be either right or obtuse, but not both.

Q: How do I classify a triangle with side lengths $8 \, \text{cm}, 15 \, \text{cm}, \text{and } 17 \, \text{cm}$?

A: To classify a triangle with side lengths $8 \, \text{cm}, 15 \, \text{cm}, \text{and } 17 \, \text{cm}$, we need to examine the side lengths and the relationships between them.

• $8^2 = 64$
• $15^2 = 225$
• $17^2 = 289$

Since $8^2 + 15^2 = 64 + 225 = 289$, the triangle is right.

Q: How do I classify a triangle with side lengths $9 \, \text{cm}, 12 \, \text{cm}, \text{and } 15 \, \text{cm}$?

A: To classify a triangle with side lengths $9 \, \text{cm}, 12 \, \text{cm}, \text{and } 15 \, \text{cm}$, we need to examine the side lengths and the relationships between them.

• $9^2 = 81$
• $12^2 = 144$
• $15^2 = 225$

Since $9^2 + 12^2 = 81 + 144 = 225$, the triangle is right.

Q: How do I classify a triangle with side lengths $10 \, \text{cm}, 13 \, \text{cm}, \text{and } 15 \, \text{cm}$?

A: To classify a triangle with side lengths $10 \, \text{cm}, 13 \, \text{cm}, \text{and } 15 \, \text{cm}$, we need to examine the side lengths and the relationships between them.

• $10^2 = 100$
• $13^2 = 169$
• $15^2 = 225$

Since $10^2 + 13^2 = 100 + 169 = 269$, the triangle is not right. Since $10^2 + 13^2 \ \textgreater \ 15^2$, the triangle is not obtuse. Therefore, the triangle is acute.

Conclusion

In conclusion, classifying triangles is a fundamental concept in mathematics. Understanding the properties of acute, right, and obtuse triangles is essential in trigonometry and geometry. By analyzing the side lengths and the relationships between them, we can determine the type of triangle. We hope this article has provided a comprehensive introduction to the classification of triangles and has helped readers understand the properties of acute, right, and obtuse triangles.