All sides are of different lengths

Oct 13, 2025 by wbrum.org 148 views

Classifying Triangles: Understanding the Properties of Acute, Right, and Obtuse Triangles

When it comes to classifying triangles based on their angles, it's essential to understand the properties of acute, right, and obtuse triangles. In this article, we'll delve into the world of triangle classification and explore the characteristics that distinguish these types of triangles.

What is a Triangle?

A triangle is a polygon with three sides and three vertices. It's a fundamental concept in geometry, and understanding the properties of triangles is crucial for solving various mathematical problems. In this article, we'll focus on classifying triangles based on their angles.

Types of Triangles

There are three main types of triangles: acute, right, and obtuse. Each type of triangle has distinct properties that can be used to classify it.

Acute Triangles

An acute triangle is a triangle with all angles less than 90 degrees. In other words, all the angles of an acute triangle are acute angles. Acute triangles have several properties that distinguish them from other types of triangles.

All angles are acute: As mentioned earlier, all the angles of an acute triangle are less than 90 degrees.
All sides are of different lengths: In an acute triangle, all three sides are of different lengths.
The sum of the squares of the two shorter sides is greater than the square of the longest side: This property is known as the triangle inequality theorem.

Right Triangles

A right triangle is a triangle with one angle equal to 90 degrees. In other words, one of the angles of a right triangle is a right angle. Right triangles have several properties that distinguish them from other types of triangles.

One angle is a right angle: As mentioned earlier, one of the angles of a right triangle is a right angle.
The square of the longest side is equal to the sum of the squares of the other two sides: This property is known as the Pythagorean theorem.
The longest side is the hypotenuse: In a right triangle, the longest side is called the hypotenuse.

Obtuse Triangles

An obtuse triangle is a triangle with one angle greater than 90 degrees. In other words, one of the angles of an obtuse triangle is an obtuse angle. Obtuse triangles have several properties that distinguish them from other types of triangles.

One angle is obtuse: As mentioned earlier, one of the angles of an obtuse triangle is an obtuse angle.
The sum of the squares of the two shorter sides is less than the square of the longest side: This property is known as the triangle inequality theorem.
The longest side is the hypotenuse: In an obtuse triangle, the longest side is called the hypotenuse.

Classifying a Triangle with Side Lengths $6 \, \text{cm}, 10 \, \text{cm},$ and $12 \, \text{cm}$

Now that we've discussed the properties of acute, right, and obtuse triangles, let's apply this knowledge to classify a triangle with side lengths $6 \, \text{cm}, 10 \, \text{cm},$ and $12 \, \text{cm}$ .

To classify this triangle, we need to determine whether it's acute, right, or obtuse. We can do this by using the triangle inequality theorem, which states that the sum of the squares of the two shorter sides is greater than the square of the longest side.

Let's calculate the sum of the squares of the two shorter sides:

6^2 + 10^2 = 36 + 100 = 136

Now, let's calculate the square of the longest side:

12^2 = 144

Since $136 \ \textless \ 144$ , we can conclude that the triangle is not acute. However, we still need to determine whether it's right or obtuse.

To do this, we can use the Pythagorean theorem, which states that the square of the longest side is equal to the sum of the squares of the other two sides. Let's calculate the sum of the squares of the other two sides:

6^2 + 10^2 = 36 + 100 = 136

Now, let's compare this value to the square of the longest side:

12^2 = 144

Since $136 \ \textless \ 144$ , we can conclude that the triangle is not right. Therefore, the triangle is obtuse.

Conclusion

In this article, we've discussed the properties of acute, right, and obtuse triangles. We've also applied this knowledge to classify a triangle with side lengths $6 \, \text{cm}, 10 \, \text{cm},$ and $12 \, \text{cm}$ . By using the triangle inequality theorem and the Pythagorean theorem, we've determined that the triangle is obtuse.

References

"Triangle Inequality Theorem." Math Open Reference, mathopenref.com/triangleinequality.html.
"Pythagorean Theorem." Math Open Reference, mathopenref.com/pythagorean.html.

Further Reading

"Geometry." Khan Academy, khanacademy.org/math/geometry.
"Triangle Classification." Math Is Fun, mathisfun.com/geometry/triangle-classification.html.
Frequently Asked Questions: Triangle Classification

In our previous article, we discussed the properties of acute, right, and obtuse triangles. We also applied this knowledge to classify a triangle with side lengths $6 \, \text{cm}, 10 \, \text{cm},$ and $12 \, \text{cm}$ . In this article, we'll answer some frequently asked questions about triangle classification.

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

A: An acute triangle is a triangle with all angles less than 90 degrees, while a right triangle is a triangle with one angle equal to 90 degrees.

Q: How can I determine whether a triangle is acute or right?

A: To determine whether a triangle is acute or right, you can use the triangle inequality theorem, which states that the sum of the squares of the two shorter sides is greater than the square of the longest side. If this condition is met, the triangle is acute. If not, it may be right.

Q: What is the Pythagorean theorem?

A: The Pythagorean theorem is a mathematical formula that states that the square of the longest side of a right triangle is equal to the sum of the squares of the other two sides.

Q: How can I use the Pythagorean theorem to classify a triangle?

A: To use the Pythagorean theorem to classify a triangle, you can calculate the sum of the squares of the two shorter sides and compare it to the square of the longest side. If the two values are equal, the triangle is right. If not, it may be acute or obtuse.

Q: What is an obtuse triangle?

A: An obtuse triangle is a triangle with one angle greater than 90 degrees.

Q: How can I determine whether a triangle is obtuse?

A: To determine whether a triangle is obtuse, you can use the triangle inequality theorem, which states that the sum of the squares of the two shorter sides is less than the square of the longest side. If this condition is met, the triangle is obtuse.

Q: Can a triangle be both acute and obtuse?

A: No, a triangle cannot be both acute and obtuse. A triangle must be either acute, right, or obtuse.

Q: Can a triangle be right and obtuse at the same time?

A: No, a triangle cannot be both right and obtuse at the same time. A triangle must be either acute, right, or obtuse.

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

A: To classify this triangle, we need to determine whether it's acute, right, or obtuse. We can do this by using the triangle inequality theorem and the Pythagorean theorem.

Let's calculate the sum of the squares of the two shorter sides:

8^2 + 15^2 = 64 + 225 = 289

Now, let's calculate the square of the longest side:

17^2 = 289

Since $289 = 289$ , we can conclude that the triangle is right.

Conclusion

In this article, we've answered some frequently asked questions about triangle classification. We've discussed the properties of acute, right, and obtuse triangles and provided examples of how to classify triangles using the triangle inequality theorem and the Pythagorean theorem.

References

"Triangle Inequality Theorem." Math Open Reference, mathopenref.com/triangleinequality.html.
"Pythagorean Theorem." Math Open Reference, mathopenref.com/pythagorean.html.

Further Reading

"Geometry." Khan Academy, khanacademy.org/math/geometry.
"Triangle Classification." Math Is Fun, mathisfun.com/geometry/triangle-classification.html.