In this great we"ll discover properties that altitudes, medians, midsegments, edge bisectors, and also perpendicular bisectors of triangles. All 4 of these types of lines or heat segments in ~ triangles room concurrent, meaning that the three medians of a triangle re-publishing intersecting points, as carry out the three altitudes, midsegments, edge bisectors, and also perpendicular bisectors. The intersecting allude is called the point of concurrency. The miscellaneous points that concurrency because that these four varieties of present or line segments all have special properties.

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Altitudes that a Triangle

The present containing the altitudes the a triangle fulfill at one allude called the orthocenter that the triangle. Since the orthocenter lies ~ above the present containing all 3 altitudes the a triangle, the segment joining the orthocenter to each side are perpendicular come the side. Keep in mind that the altitudes us aren"t have to concurrent; the lines the contain the altitudes are the only guarantee. This method that the orthocenter isn"t necessarily in the inner of the triangle.

Figure %: The currently containing the altitudes the a triangle and the orthocenter

There room two other usual theorems concerning altitudes the a triangle. Both worry the concept of similarity. The an initial states that the lengths of the altitudes of comparable triangles monitor the same proportions as the matching sides the the similar triangles.

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

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

Medians of a Triangle

Every triangle has three medians, as with it has three altitudes, angle bisectors, and also perpendicular bisectors. The medians that a triangle are the segments drawn from the vertices come the midpoints of the opposite sides. The allude of intersection that all three medians is called the centroid of the triangle. The centroid of a triangle is twice as far from a given vertex than it is native the midpoint come which the average from that vertex goes. Because that example, if a average is drawn from peak A to midpoint M v centroid C, the size of AC is twice the size of CM. The centroid is 2/3 that the method from a given vertex to the the contrary midpoint. The centroid is constantly on the inner of the triangle.

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Figure %: A triangle"s medians and centroid

Two more interesting things are true the medians. 1) The lengths that the medians of similar triangles space of the exact same proportion together the lengths of corresponding sides. 2) The average of a appropriate triangle from the appropriate angle to the hypotenuse is fifty percent the length of the hypotenuse.