What makes up an obtuse triangle

From the law of cosines , for a triangle with side lengths , , and ,. For an angle to be obtuse,. Therefore, an obtuse triangle satisfies one of , , or.

An obtuse triangle can be dissected into no fewer than seven acute triangles Wells , p. A famous problem is to find the chance that three points picked randomly in a plane are the polygon vertices of an obtuse triangle Eisenberg and Sullivan Unfortunately, the solution of the problem depends on the procedure used to pick the "random" points Portnoy In fact, it is impossible to pick random variables which are uniformly distributed in the plane Eisenberg and Sullivan Guy gives a variety of solutions to the problem.

Woolhouse solved the problem by picking uniformly distributed points in the unit disk , and obtained. The problem was generalized by Hall to -dimensional ball triangle picking , and Buchta gave closed form evaluations for Hall's integrals.

In , Lewis Carroll posed and gave another solution to the problem as follows. Call the longest side of a triangle , and call the diameter. Draw arcs from and of radius. Because the longest side of the triangle is defined to be , the third polygon vertex of the triangle must lie within the region. If the third polygon vertex lies within the semicircle , the triangle is an obtuse triangle. If the polygon vertex lies on the semicircle which will happen with probability 0 , the triangle is a right triangle.

Otherwise, it is an acute triangle. The chance of obtaining an obtuse triangle is then the ratio of the area of the semicircle to that of. The area of is then twice the area of a circular sector minus the area of the triangle. Anne Marie Helmenstine, Ph. Chemistry Expert. Helmenstine holds a Ph. She has taught science courses at the high school, college, and graduate levels. Facebook Facebook Twitter Twitter. Featured Video. Cite this Article Format. Helmenstine, Anne Marie, Ph.

Types of Triangles: Acute and Obtuse. Math Glossary: Mathematics Terms and Definitions. Math Terms: The Definition of an Angle. Angel and Angle: Commonly Confused Words. Your Privacy Rights. To change or withdraw your consent choices for ThoughtCo.

Once the height is obtained, we can find the area of an obtuse triangle by applying the formula mentioned below. The altitude or the height from the acute angles of an obtuse triangle lies outside the triangle. We extend the base as shown and determine the height of the obtuse triangle. The area of an obtuse triangle can also be found by using Heron's formula.

Each triangle has its own properties that define them. An obtuse triangle has four different properties. Let's see what they are:. Property 1: The longest side of a triangle is the side opposite to the obtuse angle.

See the image below for reference. Property 2: A triangle can only have one obtuse angle. Consider the obtuse triangle shown below. Hence, a triangle cannot have two obtuse angles because the sum of all the angles cannot exceed degrees. Observe the image given below to understand the same with an illustration. Property 4: The circumcenter and the orthocenter of an obtuse-angled triangle lie outside the triangle.

The orthocenter O , the point at which all the altitudes of a triangle intersect, lies outside in an obtuse triangle. As seen in the image below:. Circumcenter H , the median point from all the triangle vertices, lies outside in an obtuse triangle. Example 1: Which of the following angle measures can form an obtuse-angled triangle? Among the given options, option b satisfies the condition. Therefore, option b i.