Art of Problem Solving
AoPS Online
Math texts, online classes, and more
for students in grades 5-12.
Visit AoPS Online ‚
Books for Grades 5-12 Online Courses
Beast Academy
Engaging math books and online learning
for students ages 6-13.
Visit Beast Academy ‚
Books for Ages 6-13 Beast Academy Online
AoPS Academy
Small live classes for advanced math
and language arts learners in grades 2-12.
Visit AoPS Academy ‚
Find a Physical Campus Visit the Virtual Campus

Difference between revisions of "Triangle"

(→‎Definition)
m (Fixed)
 
(15 intermediate revisions by 9 users not shown)
Line 3: Line 3:
 
{{asy image|<asy>draw((0,1)--(2,0)--(3,2)--cycle);</asy>|right|A triangle.}}
 
{{asy image|<asy>draw((0,1)--(2,0)--(3,2)--cycle);</asy>|right|A triangle.}}
 
==Definition==
 
==Definition==
−
A triangle is any polygon with three [[edge | sides]], with 180 degrees. Triangles exist in Euclidean [[geometry]], and are the simplest possible polygon. In [[physics]], triangles are noted for their durability, since they have only three [[vertex|vertices]] around with to distort.
+
A '''triangle''' is any [[polygon]] with three [[edge | sides]], with the smaller angle measures of the intersections of the sides summing to 180 degrees. Triangles exist in Euclidean [[geometry]], and are the simplest possible polygon. In [[physics]], triangles are noted for their durability, since they have only three [[vertex|vertices]] around with to distort.
  
 
==Categories==
 
==Categories==
Line 10: Line 10:
 
===Equilateral===
 
===Equilateral===
 
{{main|Equilateral triangle}}
 
{{main|Equilateral triangle}}
−
An '''equilateral''' triangle has three congruent sides, and is also [[equiangular]]. Note that all equilateral triangles are [[similar]]. All the angles of equilateral triangles are <math>60^{\circ}</math>
+
An '''equilateral''' triangle has three congruent sides and is also [[equiangular]]. Note that all equilateral triangles are [[similar]]. All the angles of equilateral triangles are <math>60^{\circ}</math>
 +
 
 
===Isosceles===
 
===Isosceles===
 
{{main|Isosceles triangle}}
 
{{main|Isosceles triangle}}
−
An '''isosceles''' triangle has at least two congruent sides (this means that all equilateral triangles are also isosceles), and the two angles opposite the congruent sides are also congruent (this is commonly known as the base angles theorem).
+
An '''isosceles''' triangle has at least two congruent sides (this means that all equilateral triangles are also isosceles), and the two angles opposite the congruent sides are also congruent (this is commonly known as the [[Hinge theorem]]).
 +
 
 
===Scalene===
 
===Scalene===
 
{{main|Scalene triangle}}
 
{{main|Scalene triangle}}
−
A '''scalene''' triangle has no congruent sides, and also no congruent angles (this can be proven by the converse of the [[Hinge Theorem]].
+
A '''scalene''' triangle has no congruent sides, and also no congruent angles (this can be proven by the converse of the Hinge Theorem).
 +
 
 
===Right===
 
===Right===
 
{{asy image|<asy>draw((0,0)--(1,0)--(0,1)--cycle);</asy>|right|A right triangle.}}
 
{{asy image|<asy>draw((0,0)--(1,0)--(0,1)--cycle);</asy>|right|A right triangle.}}
Line 25: Line 28:
 
===Acute===
 
===Acute===
 
All the angles of an '''acute''' triangle are [[acute angle]]s.
 
All the angles of an '''acute''' triangle are [[acute angle]]s.
−
==Related Formulae and Theorems==
+
 
 +
==Related Formulas and Theorems==
 
*The [[area]] of any triangle with base <math>b</math> and height <math>h</math> is <math>\frac{bh}{2}</math>. (This can be shown by combining the triangle and a copy of it into a [[parallelogram]]).
 
*The [[area]] of any triangle with base <math>b</math> and height <math>h</math> is <math>\frac{bh}{2}</math>. (This can be shown by combining the triangle and a copy of it into a [[parallelogram]]).
−
*The [[area]] of any triangle with sides <math>a,b,c</math> opposite angles <math>A,B,C</math> is  
+
*The [[area]] of any triangle with sides <math>a,b,c</math> opposite angles <math>A,B,C</math> is <math>\frac {ab}{2}\sin C </math>
−
<math>\frac {ab}{2}\sin C </math>
+
*The area of any triangle with sides <math>a,b,c</math> is <math>\sqrt{s(s-a)(s-b)(s-c)}</math>, where <math>s</math> is the [[semiperimeter]] ([[Heron's Formula]]).
 
*For a right triangle with legs <math>a,b</math> and [[hypotenuse]] <math>c</math>, <math>a^2+b^2=c^2</math>. This is the famous [[Pythagorean theorem]].
 
*For a right triangle with legs <math>a,b</math> and [[hypotenuse]] <math>c</math>, <math>a^2+b^2=c^2</math>. This is the famous [[Pythagorean theorem]].
 
*The [[inradius]] of a triangle with sides <math>a,b,c</math> and area <math>K</math> is <math>\frac{2K}{a+b+c}</math>.
 
*The [[inradius]] of a triangle with sides <math>a,b,c</math> and area <math>K</math> is <math>\frac{2K}{a+b+c}</math>.
 +
*In any triangle, the sum of any two sides is greater than the length of the third side.
 
*The sum of the interior angles of a triangle is <math>180^{\circ}</math>.
 
*The sum of the interior angles of a triangle is <math>180^{\circ}</math>.
 
*See [[trigonometric identities]] for a list of formulae related to trigonometry.
 
*See [[trigonometric identities]] for a list of formulae related to trigonometry.
Line 38: Line 43:
 
* [http://www.amazon.com/exec/obidos/ASIN/0486691543/artofproblems-20 Challenging Problems in Geometry] - A good book for students who already have a solid handle on elementary geometry.
 
* [http://www.amazon.com/exec/obidos/ASIN/0486691543/artofproblems-20 Challenging Problems in Geometry] - A good book for students who already have a solid handle on elementary geometry.
 
* [http://www.amazon.com/exec/obidos/ASIN/0883856190/artofproblems-20 Geometry Revisited] - A classic.
 
* [http://www.amazon.com/exec/obidos/ASIN/0883856190/artofproblems-20 Geometry Revisited] - A classic.
 +
* [http://www.ams.org/journals/bull/1902-08-10/S0002-9904-1902-00923-3/S0002-9904-1902-00923-3.pdf Mathematical Problems] - Lecture delivered before the International Congress of Mathematicians at Paris in 1900 by David Hilbert.
 +
* [http://www.gutenberg.org/files/17384/17384-pdf.pdf Foundations of Geometry] by David Hilbert
  
 
==See Also==
 
==See Also==

Latest revision as of 20:16, 8 October 2024

A triangle is a type of polygon.

[asy]draw((0,1)--(2,0)--(3,2)--cycle);[/asy]

Enlarge.png
A triangle.

Definition

A triangle is any polygon with three sides, with the smaller angle measures of the intersections of the sides summing to 180 degrees. Triangles exist in Euclidean geometry, and are the simplest possible polygon. In physics, triangles are noted for their durability, since they have only three vertices around with to distort.

Categories

Triangles are split into six categories; three by their angles and three by their side lengths.

[asy]draw((0,0)--(1,0)--(0.5,0.5)--cycle);[/asy]

Enlarge.png
An isosceles triangle.

Equilateral

Main article: Equilateral triangle

An equilateral triangle has three congruent sides and is also equiangular. Note that all equilateral triangles are similar. All the angles of equilateral triangles are $60^{\circ}$

Isosceles

Main article: Isosceles triangle

An isosceles triangle has at least two congruent sides (this means that all equilateral triangles are also isosceles), and the two angles opposite the congruent sides are also congruent (this is commonly known as the Hinge theorem).

Scalene

Main article: Scalene triangle

A scalene triangle has no congruent sides, and also no congruent angles (this can be proven by the converse of the Hinge Theorem).

Right

[asy]draw((0,0)--(1,0)--(0,1)--cycle);[/asy]

Enlarge.png
A right triangle.
Main article: Right triangle

A right triangle has a right angle, which means the other two angles are complementary. Trigonometry is largely based on right triangles, and the famous Pythagorean Theorem deals with the side lengths of the right triangle. Note that it is impossible to have an equilateral right triangle. It is also impossible to have more than one right angle in a triangle.

Obtuse

An obtuse triangle has an obtuse angle. Note that it is impossible to have an equilateral obtuse triangle. It is also impossible to have more than one obtuse angle in a triangle.

Acute

All the angles of an acute triangle are acute angles.

Related Formulas and Theorems

  • The area of any triangle with base $b$ and height $h$ is $\frac{bh}{2}$. (This can be shown by combining the triangle and a copy of it into a parallelogram).
  • The area of any triangle with sides $a,b,c$ opposite angles $A,B,C$ is $\frac {ab}{2}\sin C$
  • The area of any triangle with sides $a,b,c$ is $\sqrt{s(s-a)(s-b)(s-c)}$, where $s$ is the semiperimeter (Heron's Formula).
  • For a right triangle with legs $a,b$ and hypotenuse $c$, $a^2+b^2=c^2$. This is the famous Pythagorean theorem.
  • The inradius of a triangle with sides $a,b,c$ and area $K$ is $\frac{2K}{a+b+c}$.
  • In any triangle, the sum of any two sides is greater than the length of the third side.
  • The sum of the interior angles of a triangle is $180^{\circ}$.
  • See trigonometric identities for a list of formulae related to trigonometry.

External Links

See Also

Retrieved from "https://artofproblemsolving.com/wiki/index.php?title=Triangle&oldid=229344"