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Discuss the following properties of a triangle: The altitudes of a triangle are concurrent at a point. This point is called the orthocenter of the given triangle.

Sagot :

First, the definitions:
Altitude - is a segment drawn from a vertex perpendicular to the opposite side. Three altitudes can be drawn in a triangle.

Concurrent - describes the three or more lines that intersect at a point.

Point of Concurrency - the point where the lines or segments intersect.

Orthocenter - is the point containing the altitudes that are concurrent.  It is the point of concurrency of the concurrent altitudes.  "Ortho" is a prefix than means right, because each line that intersects at this point is  perpendicular to a corresponding side in a triangle.  This characteristic of orthocenter makes it different from                  circumcenter and centroid.

From the definitions above, you can derive the following statements:
1)  The altitudes (three) of a triangle are concurrent at a point.  It means that the the tree altitudes intersect at a point, which is called orthocenter (the point of concurrency)

2)  The lines containing the altitudes are concurrent at a point.

3)  The orhocenter of an acute triangle is inside the given traingle.

4)  The orthocenter of an obtuse triangle is outside the triangle.

5)  The orthocenter of a given triangle is not necessarily inside the triangle.  

(Please practice drawing different kinds of triangles.  It will help you in understanding their properties and characteristics.  Eventually, you'd be able to write proofs and reasons.)
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