In this great we"ll learn properties that altitudes, medians, midsegments, edge bisectors, and also perpendicular bisectors that triangles. All four of these varieties of present or line segments within triangles space concurrent, definition that the three medians the a triangle re-publishing intersecting points, as perform the 3 altitudes, midsegments, angle bisectors, and also perpendicular bisectors. The intersecting point is dubbed the point of concurrency. The assorted points of concurrency for these four species of currently or line segments all have actually special properties.

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### Altitudes of a Triangle

The currently containing the altitudes that a triangle fulfill at one suggest called the orthocenter of the triangle. Since the orthocenter lies top top the lines containing all 3 altitudes the a triangle, the segment joining the orthocenter to every side are perpendicular come the side. Save in mind the the altitudes us aren"t necessarily concurrent; the lines the contain the altitudes are the only guarantee. This way that the orthocenter isn"t have to in the interior of the triangle.

Figure %: The present containing the altitudes of a triangle and also the orthocenter

There space two other usual theorems worrying altitudes of a triangle. Both problem the ide of similarity. The very first states the the lengths the the altitudes of comparable triangles follow the exact same proportions together the matching sides that the similar triangles.

The second states the the altitude the a right triangle drawn from the appropriate angle to the hypotenuse divides the triangle right into two similar triangles. These 2 triangles are also similar to the initial triangle. The figure listed below illustrates this concept.

Figure %: triangles ABC, DAC, and DBA are similar to one another

### Medians of a Triangle

Every triangle has three medians, similar to it has three altitudes, angle bisectors, and perpendicular bisectors. The medians the a triangle are the segments drawn from the vertices come the midpoints of the opposite sides. The suggest of intersection the all three medians is dubbed the centroid of the triangle. The centroid that a triangle is double as much from a offered vertex 보다 it is indigenous the midpoint come which the median from that vertex goes. For example, if a typical is drawn from crest A to midpoint M with centroid C, the length of AC is double the length of CM. The centroid is 2/3 of the method from a offered vertex come the the contrary midpoint. The centroid is constantly on the interior of the triangle.

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Figure %: A triangle"s medians and also centroid

Two much more interesting things are true that medians. 1) The lengths that the medians of similar triangles room of the exact same proportion as the lengths of matching sides. 2) The typical of a best triangle from the appropriate angle to the hypotenuse is fifty percent the size of the hypotenuse.