Difference between revisions of "Altitude"
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| + | <i>The altitude defined is as in a mathematical sense, there are more definitions of the word altitude not mentioned here</i> | ||
| + | ==Definition== | ||
| + | |||
In [[geometry]], an '''altitude''' of a figure is a [[cevian]] that is [[perpendicular]] to the side to which it extends. | In [[geometry]], an '''altitude''' of a figure is a [[cevian]] that is [[perpendicular]] to the side to which it extends. | ||
Usually, one is concerned with the altitude (or ''height'') of [[triangle]]s. In particular, the altitudes of any triangle are [[concurrent]] at a point known as the [[orthocenter]]. | Usually, one is concerned with the altitude (or ''height'') of [[triangle]]s. In particular, the altitudes of any triangle are [[concurrent]] at a point known as the [[orthocenter]]. | ||
| + | |||
| + | ==Usage== | ||
| + | |||
| + | Altitudes are mainly used to finding the [[area]] of a triangle, frequently used as the height in <math>\frac{1}{2} bh</math>. Furthermore, if one knows the area of a triangle and one side, the altitude to that side can be calculated. In addition to triangles, altitudes are also useful for finding the area of quadrilaterals with at least one pair of parallel sides (rectangles, parallelograms, trapezoids). | ||
| + | |||
| + | Because the altitude is always perpendicular to a side, creating right angles, many right triangle tools like the [[Pythagorean Theorem]] can be used and can be very useful. | ||
| + | |||
| + | ==Diagram== | ||
| + | <asy> | ||
| + | size(15cm); | ||
| + | markscalefactor = 0.01; | ||
| + | dot((0,0)); | ||
| + | dot((4,0)); | ||
| + | dot((3,3)); | ||
| + | dot((3,0)); | ||
| + | dot((2,2)); | ||
| + | dot((3,1)); | ||
| + | dot((3.6,1.2)); | ||
| + | draw((0,0)--(4,0)--(3,3)--cycle); | ||
| + | draw((0,0)--(3.6,1.2),red); | ||
| + | draw((2,2)--(4,0),red); | ||
| + | draw((3,0)--(3,3), red); | ||
| + | draw(rightanglemark((0,0),(3,0),(3,1))); | ||
| + | draw(rightanglemark((4,0),(3.6,1.2),(0,0))); | ||
| + | draw(rightanglemark((0,0),(2,2),(4,0))); | ||
| + | label("Orthocenter",(2.85,1.1),W); | ||
| + | </asy> | ||
| + | The lines highlighted are the altitudes of the triangle, they meet at the [[orthocenter]]. | ||
==See also== | ==See also== | ||
*[[Geometry]] | *[[Geometry]] | ||
*[[Area]] | *[[Area]] | ||
| + | *[[Orthocenter]] | ||
[[Category:Geometry]] | [[Category:Geometry]] | ||
{{stub}} | {{stub}} | ||
| − | |||
| − | |||
Revision as of 21:00, 16 January 2020
The altitude defined is as in a mathematical sense, there are more definitions of the word altitude not mentioned here
Contents
Definition
In geometry, an altitude of a figure is a cevian that is perpendicular to the side to which it extends.
Usually, one is concerned with the altitude (or height) of triangles. In particular, the altitudes of any triangle are concurrent at a point known as the orthocenter.
Usage
Altitudes are mainly used to finding the area of a triangle, frequently used as the height in 
. Furthermore, if one knows the area of a triangle and one side, the altitude to that side can be calculated. In addition to triangles, altitudes are also useful for finding the area of quadrilaterals with at least one pair of parallel sides (rectangles, parallelograms, trapezoids).
Because the altitude is always perpendicular to a side, creating right angles, many right triangle tools like the Pythagorean Theorem can be used and can be very useful.
Diagram
![[asy] size(15cm); markscalefactor = 0.01; dot((0,0)); dot((4,0)); dot((3,3)); dot((3,0)); dot((2,2)); dot((3,1)); dot((3.6,1.2)); draw((0,0)--(4,0)--(3,3)--cycle); draw((0,0)--(3.6,1.2),red); draw((2,2)--(4,0),red); draw((3,0)--(3,3), red); draw(rightanglemark((0,0),(3,0),(3,1))); draw(rightanglemark((4,0),(3.6,1.2),(0,0))); draw(rightanglemark((0,0),(2,2),(4,0))); label("Orthocenter",(2.85,1.1),W); [/asy]](http://latex.artofproblemsolving.com/3/b/4/3b484374a3d190b7da30ce26d184e6ae3305e73c.png)
The lines highlighted are the altitudes of the triangle, they meet at the orthocenter.
See also
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