Difference between revisions of "Acute triangle"
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* A triangle is acute if and only if each of its [[altitude]]s lies entirely in the triangle's interior. | * A triangle is acute if and only if each of its [[altitude]]s lies entirely in the triangle's interior. | ||
| − | * A triangle with sides of length <math>a, b</math> and <math>c</math> is acute if and only if <math>a^2 + b^2 > c^2</math>, <math>b^2 + c^2 > a^2</math> and <math>c^2 + a^2 > b^2</math>. This is known as the [[Geometric inequality | Pythagorean Inequality]]. It is a consequence of the Law of Cosines as the cosine of an angle that is less than 90 degrees is positive | + | * A triangle with sides of length <math>a, b</math> and <math>c</math> is acute if and only if <math>a^2 + b^2 > c^2</math>, <math>b^2 + c^2 > a^2</math> and <math>c^2 + a^2 > b^2</math>. This is known as the [[Geometric inequality | Pythagorean Inequality]]. It is a consequence of the Law of Cosines as the cosine of an angle that is less than 90 degrees is positive. |
In an acute triangle, the [[circumcenter]], [[incenter]], [[orthocenter]], and [[centroid]] are all within the interior of the triangle. | In an acute triangle, the [[circumcenter]], [[incenter]], [[orthocenter]], and [[centroid]] are all within the interior of the triangle. | ||
Latest revision as of 12:09, 20 February 2024
An acute triangle is a triangle in which each angle is an acute angle. Any triangle which is not acute is either a right triangle or an obtuse triangle.
All acute triangle angles are less than 90 degrees. For example, an equilateral triangle is always acute, since all angles (which are 60) are all less than 90.
Acute triangles can also be defined in different ways:
- A triangle is acute if and only if each of its altitudes lies entirely in the triangle's interior.
- A triangle with sides of length
and 
is acute if and only if 
, 
and 
. This is known as the Pythagorean Inequality. It is a consequence of the Law of Cosines as the cosine of an angle that is less than 90 degrees is positive.
and 
is acute if and only if 
, 
and 
. This is known as the In an acute triangle, the circumcenter, incenter, orthocenter, and centroid are all within the interior of the triangle.
An Acute triangle can be an equilateral triangle and also can be scalene. There are Acute triangle,Right triangles, and Obtuse triangles.
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