Difference between revisions of "Obtuse triangle"
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| − | + | An '''obtuse triangle''' is a [[triangle]] in which one [[angle]] is an [[obtuse angle]]. Any triangle which is not obtuse is either a [[right triangle]] or an [[acute triangle]]. | |
| + | |||
| + | <asy> | ||
| + | unitsize(1cm); | ||
| + | pair A, B, C, D, E, F; | ||
| + | |||
| + | A = (0, 0); | ||
| + | B = (2, 0); | ||
| + | C = (-1, 4); | ||
| + | |||
| + | draw(A--B--C--cycle); | ||
| + | draw(anglemark(B, A, C, 10)); | ||
| + | label("$100^\circ$", A, 3NE); | ||
| + | |||
| + | D = (3, 0); | ||
| + | E = (7, 0); | ||
| + | F = (5, 2); | ||
| − | + | draw(D--E--F--cycle); | |
| + | draw(anglemark(D, F, E, 10)); | ||
| + | label("$140^\circ$", F, 4S); | ||
| + | </asy> | ||
| + | Obtuse triangles only have one obtuse angle because the angles of a triangle must sum to <math>180^\circ</math>. Two obtuse angles would sum to more than that. | ||
| − | + | Obtuse triangles can be defined in other ways: | |
* A triangle is obtuse if and only if two of its [[altitude]]s lie entirely outside the triangle. (There is no triangle with exactly one altitude or all three altitudes outside the triangle.) | * A triangle is obtuse if and only if two of its [[altitude]]s lie entirely outside the triangle. (There is no triangle with exactly one altitude or all three altitudes outside the triangle.) | ||
| + | * A triangle with sides of length <math>a</math>, <math>b</math>, and <math>c</math> where <math>c > a, b</math>, is obtuse if and only if <math>a^2 + b^2 < c^2</math>. This is known as the [[Geometric inequality | Pythagorean Inequality]]. It follows directly from the [[Law of Cosines]]. | ||
| + | {{stub}} | ||
| − | + | [[Category:Definition]] | |
| + | [[Category:Geometry]] | ||
Latest revision as of 13:20, 12 June 2022
An obtuse triangle is a triangle in which one angle is an obtuse angle. Any triangle which is not obtuse is either a right triangle or an acute triangle.
![[asy] unitsize(1cm); pair A, B, C, D, E, F; A = (0, 0); B = (2, 0); C = (-1, 4); draw(A--B--C--cycle); draw(anglemark(B, A, C, 10)); label("$100^\circ$", A, 3NE); D = (3, 0); E = (7, 0); F = (5, 2); draw(D--E--F--cycle); draw(anglemark(D, F, E, 10)); label("$140^\circ$", F, 4S); [/asy]](http://latex.artofproblemsolving.com/7/a/8/7a8842c5da6df412c69e32092fff83de88d03c4d.png)
Obtuse triangles only have one obtuse angle because the angles of a triangle must sum to 
. Two obtuse angles would sum to more than that.
Obtuse triangles can be defined in other ways:
- A triangle is obtuse if and only if two of its altitudes lie entirely outside the triangle. (There is no triangle with exactly one altitude or all three altitudes outside the triangle.)
- A triangle with sides of length
, 
, and 
where 
, is obtuse if and only if 
. This is known as the Pythagorean Inequality. It follows directly from the Law of Cosines.
, 
, and 
where 
, is obtuse if and only if 
. This is known as the This article is a stub. Help us out by expanding it.
