# Are Random Triangles Acute or Obtuse?

Gilbert Strang
Professor of Mathematics
Massachusetts Institute of Technology
Cambridge, Massachusetts 02139 USA

This learning video deals with a question of geometrical probability. A key idea presented is the fact that a linear equation in three dimensions produces a plane. The video focuses on random triangles that are defined by their three respective angles. These angles are chosen randomly subject to a constraint that they must sum to 180 degrees. One class period is required to complete this learning video, and the only prerequisites are a familiarity with geometry and an understanding of the equation for a plane, which is presented in the module. Materials needed for this lesson include blackboard and chalk. Optional materials include a cardboard box and colored paper. An example of the types of in-class activities for between segments of the video is: Ask six students for numbers and make those numbers the coordinates x,y of three points. Then have the class try to figure out how to decide if the triangle with those corners is acute or obtuse.

Professor Strang teaches Linear Algebra and Computational Science at MIT, and both of these classes are videotaped and available on MIT's OpenCourseWare ocw.mit.edu. He also writes research papers and textbooks on these subjects. Click here to read more about Professor Strang.

Professor Strang’s Linear Algebra Class lecture videos
http://web.mit.edu/18.06/www/Video/video-fall-99.html

An online discussion of the problem discussed in this learning video
http://mathworld.wolfram.com/ObtuseTriangle.html

A free interactive math textbook on the web. Initially covering high-school geometry
http://www.mathopenref.com/obtusetriangle.html

An interactive lesson connecting probability and geometry
http://www.shodor.org/interactivate/lessons/ProbabilityGeometry/

Provides extensive resources for the study of geometry
http://mathworld.wolfram.com/topics/Geometry.html

#### An obtuse triangle is a

November 8, 2011 at 1:19 am

An obtuse triangle is a triangle in which one of the angles is an obtuse angle. (Obviously, only a single angle in a triangle can be obtuse or it wouldn't be a triangle.) A triangle must be either obtuse, acute, or right.

From the law of cosines, for a triangle with side lengths , , and ,

(1)
with the angle opposite side . 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 1986, p. 71).

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 1996). Unfortunately, the solution of the problem depends on the procedure used to pick the "random" points (Portnoy 1994). In fact, it is impossible to pick random variables which are uniformly distributed in the plane (Eisenberg and Sullivan 1996). Guy (1993) gives a variety of solutions to the problem. Woolhouse (1886) solved the problem by picking uniformly distributed points in the unit disk, and obtained
Gilbert Strange had explained the whole in a ver simple words .These all can be found on online tutorial site including online solver

This Lesson is in the following clusters: Geometry, Probability

#### An obtuse triangle is a

November 8, 2011 at 1:19 am

An obtuse triangle is a triangle in which one of the angles is an obtuse angle. (Obviously, only a single angle in a triangle can be obtuse or it wouldn't be a triangle.) A triangle must be either obtuse, acute, or right.

From the law of cosines, for a triangle with side lengths , , and ,

(1)
with the angle opposite side . 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 1986, p. 71).

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 1996). Unfortunately, the solution of the problem depends on the procedure used to pick the "random" points (Portnoy 1994). In fact, it is impossible to pick random variables which are uniformly distributed in the plane (Eisenberg and Sullivan 1996). Guy (1993) gives a variety of solutions to the problem. Woolhouse (1886) solved the problem by picking uniformly distributed points in the unit disk, and obtained
Gilbert Strange had explained the whole in a ver simple words .These all can be found on online tutorial site including online solver