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Geometry: Theorems

Theorems for Segments within Triangles

Problems

Theorems for Segments within Triangles, page 2

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In this lesson we'll learn properties of altitudes, medians, midsegments, angle bisectors, and perpendicular bisectors of triangles. All four of these types of lines or line segments within triangles are concurrent, meaning that the three medians of a triangle share intersecting points, as do the three altitudes, midsegments, angle bisectors, and perpendicular bisectors. The intersecting point is called the point of concurrency. The various points of concurrency for these four types of lines or line segments all have special properties.

Altitudes of a Triangle

The lines containing the altitudes of a triangle meet at one point called the orthocenter of the triangle. Because the orthocenter lies on the lines containing all three altitudes of a triangle, the segments joining the orthocenter to each side are perpendicular to the side. Keep in mind that the altitudes themselves aren't necessarily concurrent; the lines that contain the altitudes are the only guarantee. This means that the orthocenter isn't necessarily in the interior of the triangle.

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

There are two other common theorems concerning altitudes of a triangle. Both concern the concept of similarity. The first states that the lengths of the altitudes of similar triangles follow the same proportions as the corresponding sides of the similar triangles.

The second states that the altitude of a right triangle drawn from the right angle to the hypotenuse divides the triangle into two similar triangles. These two triangles are also similar to the original 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, just like it has three altitudes, angle bisectors, and perpendicular bisectors. The medians of a triangle are the segments drawn from the vertices to the midpoints of the opposite sides. The point of intersection of 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 from the midpoint to which the median from that vertex goes. For example, if a median is drawn from vertex A to midpoint M through centroid C, the length of AC is twice the length of CM. The centroid is 2/3 of the way from a given vertex to the opposite midpoint. The centroid is always on the interior of the triangle.

Figure %: A triangle's medians and centroid

Two more interesting things are true of medians. 1) The lengths of the medians of similar triangles are of the same proportion as the lengths of corresponding sides. 2) The median of a right triangle from the right angle to the hypotenuse is half the length of the hypotenuse.

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