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Obtuse Triangle – Definition with Examples

What is an Obtuse Triangle?

An obtuse triangle is a type of triangle where one of its angles measures more than 90 degrees. These triangles are unique and fascinating, as they possess special characteristics that distinguish them from other types of triangles, such as acute and right triangles. The study of obtuse triangles can help students develop a deeper understanding of the principles of geometry and trigonometry.

In this article, we will explore various aspects of obtuse triangles, including their types, properties, and formulas. We will also provide some solved examples and practice problems to help you enhance your understanding of this intriguing triangle.

Types of Obtuse Triangles

There are three main types of obtuse triangles, based on the length of their sides:

1. Obtuse-angled scalene triangle: In this type of obtuse triangle, all three sides have different lengths. As a result, all three angles are also different.

2. Obtuse-angled isosceles triangle: This triangle has two sides of equal length, and therefore, two angles are equal. However, one angle remains obtuse.

3. Obtuse-angled equilateral triangle: This triangle doesn’t exist. An equilateral triangle has all sides equal and all angles measuring 60 degrees, making it an acute triangle.

Properties of Obtuse Triangles

Obtuse triangles have several interesting properties that distinguish them from other triangles:

1. In an obtuse triangle, there is always one angle measuring greater than 90 degrees.

2. The side opposite the obtuse angle is the longest side of the triangle.

3. The sum of the other two angles in an obtuse triangle is always less than 90 degrees.

4. The altitude drawn from the obtuse angle in an obtuse triangle lies outside the triangle.

5. An obtuse triangle can never be a right triangle or an equilateral triangle.

Special Facts About the Obtuse Triangle

There are a few special facts about obtuse triangles that make them stand out from other triangles:

1. The circumcenter (the point where the perpendicular bisectors of the sides meet) of an obtuse triangle always lies outside the triangle.

2. The orthocenter (the point where the altitudes of the triangle intersect) is also located outside the triangle in an obtuse triangle.

3. The incenter (the point where angle bisectors intersect) is the only center that lies inside an obtuse triangle.

4. The longest side of an obtuse triangle is always opposite the obtuse angle.

5. The obtuse angle in an obtuse triangle cannot be bisected into two acute angles.

Obtuse Angled Triangle Formula

The formula to find the angles of an obtuse triangle is the same as that for any other triangle. According to the angle sum property of a triangle, the sum of all angles in a triangle always equals 180 degrees. If one angle is obtuse, we can calculate the remaining angles using this formula:

angle1 + angle2 + angle3 = 180°

Formula for the Perimeter of an Obtuse Triangle

The formula for calculating the perimeter of an obtuse triangle is the same as for any other triangle:

Perimeter = side1 + side2 + side3

Simply add the lengths of all three sides to find the perimeter of the obtuse triangle.

Formula for the Area of an Obtuse Triangle

To calculate the area of an obtuse triangle, you can use Heron’s formula, which works for any type of triangle:

Area = √(s*(s-side1)(s-side2)(s-side3))

Where s is the semi-perimeter, calculated as:

s = (side1 + side2 + side3) / 2

Solved Examples

Here are some solved examples to help you better understand obtuse triangles and their properties:

Example 1: If an obtuse triangle has sides measuring 5, 12, and 14 units, find the obtuse angle.

Solution: To find the obtuse angle, we can use the Law of Cosines. Let the obtuse angle be between sides 12 and 14.

cos(angle) = (side1^2 + side2^2 - side3^2) / (2*side1*side2)

cos(angle) = (12^2 + 14^2 - 5^2) / (2*12*14)

cos(angle) = (144 + 196 - 25) / (336)

cos(angle) = 0.476190476

Now, find the angle using the inverse cosine function:

angle = arccos(0.476190476)

angle ≈ 61.93°

Since the other two angles’ sum must be less than 90°, we can conclude that 61.93° is the obtuse angle in this triangle.

Example 2: Calculate the area of an obtuse triangle with sides measuring 7, 15, and 18 units.

Solution: Use Heron’s formula to calculate the area.

First, find the semi-perimeter s:

s = (7 + 15 + 18) / 2 = 20

Now, apply Heron’s formula:

Area = √(20*(20-7)*(20-15)*(20-18))

Area = √(20*13*5*2)

Area ≈ 52.2 square units

Practice Problems

1. If an obtuse triangle has angles measuring 120°, 30°, and x°, find the value of x.

2. Calculate the perimeter of an obtuse triangle with sides measuring 8, 15, and 20 units.

3. Find the area of an obtuse triangle with sides measuring 9, 12, and 15 units.

Conclusion

We hope this in-depth article has not only enriched your understanding of obtuse triangles, their properties, and their applications, but also sparked your curiosity to explore more about the world of geometry and trigonometry. Obtuse triangles are an essential part of these fields, and mastering them can pave the way to a deeper appreciation of mathematics.

At Brighterly, we are committed to helping young learners excel in their mathematical journey, and we believe that a strong foundation in geometry, including the study of obtuse triangles, plays a critical role in achieving this goal. Our dedicated team of experts continually strives to create unique, high-quality, and engaging content that caters to the diverse learning needs of children.

So, keep practicing, exploring, and discovering the fascinating world of mathematics with Brighterly. Don’t forget to check out our other articles and resources to nurture your curiosity and expand your knowledge. Happy learning, and remember, with Brighterly!

Frequently Asked Questions on Obtuse Triangle

What is an obtuse triangle?

An obtuse triangle is a unique type of triangle characterized by having one angle that measures more than 90 degrees. This distinctive feature sets obtuse triangles apart from other triangle categories such as acute triangles (all angles less than 90 degrees) and right triangles (one angle equal to 90 degrees). Obtuse triangles are essential in the study of geometry and trigonometry and can provide valuable insights into the properties and relationships within triangles.

Can an obtuse triangle be equilateral?

No, an obtuse triangle cannot be equilateral. An equilateral triangle has all three sides equal in length, which results in all three angles measuring 60 degrees each. Therefore, equilateral triangles are a type of acute triangle, as all angles are less than 90 degrees. Obtuse triangles must have one angle larger than 90 degrees, making it impossible for them to be equilateral.

What is the sum of angles in an obtuse triangle?

The sum of angles in any triangle, including obtuse triangles, is always 180 degrees. This is a fundamental property of all triangles, regardless of their shape or size. In an obtuse triangle, one angle is greater than 90 degrees, while the other two angles must be acute (less than 90 degrees) to ensure that their sum equals 180 degrees.

Can an obtuse triangle have two obtuse angles?

No, an obtuse triangle can have only one obtuse angle. If a triangle were to have two obtuse angles, the sum of its angles would exceed 180 degrees, violating the fundamental property of triangles, which states that the sum of all angles in a triangle must always equal 180 degrees. In an obtuse triangle, the other two angles are acute, ensuring that their sum, along with the obtuse angle, adds up to precisely 180 degrees.

Information Sources

1. Obtuse Triangle – Wikipedia
2. Obtuse Triangle – Brilliant Wiki
3. Understanding Obtuse Triangles – Universal Class