obtuse triangle

NOVEMBER 14, 2023

Obtuse Triangle: Definition and Properties

Definition

An obtuse triangle is a type of triangle in geometry where one of the angles measures more than 90 degrees. In other words, it is a triangle that has one angle greater than a right angle.

History

The concept of obtuse triangles has been studied for centuries. Ancient Greek mathematicians, such as Euclid, recognized and explored the properties of different types of triangles, including obtuse triangles.

Grade Level

The concept of obtuse triangles is typically introduced in middle school mathematics, around grades 6-8. Students learn about different types of triangles and their properties, including obtuse triangles.

Knowledge Points

To understand obtuse triangles, students should have a basic understanding of angles, particularly acute angles (less than 90 degrees) and right angles (90 degrees). They should also be familiar with the properties of triangles, such as the sum of angles in a triangle being 180 degrees.

Types of Obtuse Triangles

There are no specific types of obtuse triangles based on side lengths or other characteristics. The classification of an obtuse triangle is solely based on the measure of one of its angles.

Properties of Obtuse Triangles

  1. An obtuse triangle has one angle greater than 90 degrees.
  2. The sum of the measures of the three angles in an obtuse triangle is always 180 degrees.
  3. The other two angles in an obtuse triangle are acute angles (less than 90 degrees).
  4. The longest side of an obtuse triangle is opposite the obtuse angle.

Finding or Calculating an Obtuse Triangle

To find or calculate an obtuse triangle, you need to know at least one angle measure and the lengths of at least two sides. With this information, you can use trigonometric functions, such as sine, cosine, and tangent, to find the remaining side lengths and angle measures.

Formula or Equation for Obtuse Triangle

There is no specific formula or equation exclusively for obtuse triangles. However, the general formulas and equations for triangles, such as the Law of Sines and the Law of Cosines, can be used to solve problems involving obtuse triangles.

Applying the Obtuse Triangle Formula or Equation

When using the general formulas and equations for triangles, you can apply them to obtuse triangles by considering the given angle measures and side lengths. By substituting the known values into the appropriate formulas, you can solve for the unknowns.

Symbol or Abbreviation for Obtuse Triangle

There is no specific symbol or abbreviation for an obtuse triangle. It is commonly referred to as an "obtuse triangle" or simply described as a triangle with one obtuse angle.

Methods for Obtuse Triangle

To solve problems involving obtuse triangles, you can use various methods, including trigonometry, the Pythagorean theorem, and the properties of triangles. These methods allow you to find missing side lengths, angle measures, or other information about the triangle.

Solved Examples on Obtuse Triangle

  1. Given an obtuse triangle with angle measures of 100 degrees, 30 degrees, and 50 degrees, find the length of the longest side.
  2. In an obtuse triangle, if the lengths of two sides are 5 cm and 8 cm, and the included angle is 120 degrees, find the length of the third side.
  3. An obtuse triangle has side lengths of 7 cm, 9 cm, and 12 cm. Find the measure of the smallest angle.

Practice Problems on Obtuse Triangle

  1. Find the missing angle in an obtuse triangle with angle measures of 110 degrees and 20 degrees.
  2. In an obtuse triangle, if the lengths of two sides are 6 cm and 10 cm, and the included angle is 150 degrees, find the length of the third side.
  3. An obtuse triangle has side lengths of 8 cm, 10 cm, and 15 cm. Find the measure of the largest angle.

FAQ on Obtuse Triangle

Q: What is an obtuse triangle? A: An obtuse triangle is a triangle with one angle greater than 90 degrees.

Q: How do you find the longest side in an obtuse triangle? A: The longest side in an obtuse triangle is opposite the obtuse angle.

Q: Can an obtuse triangle have two obtuse angles? A: No, an obtuse triangle can only have one obtuse angle. The other two angles must be acute.

Q: What is the sum of the angles in an obtuse triangle? A: The sum of the measures of the three angles in an obtuse triangle is always 180 degrees.

Q: Can an obtuse triangle be equilateral? A: No, an equilateral triangle has three equal angles of 60 degrees each, which are all acute angles. Therefore, an obtuse triangle cannot be equilateral.