5.1.2 Medians, Altitudes, and Midsegments in Triangles (continued)
Example 3 Finding the Equation of an Altitude
An altitude of a triangle is a perpendicular segment from a vertex to the line containing the opposite side. So, an altitude is similar to a median since they both pass through a vertex and the opposite side. However, the median must pass through the midpoint of that opposite side, but the altitude does not necessarily pass through the midpoint of the opposite side. The altitude passes through the point which makes the segment perpendicular to the opposite side. The altitude and median are the same line when that line is the perpendicular bisector of the side opposite of the vertex. As in the previous example, the coordinates of the vertex C , ( − 4, − 1), can be used to write the equation of the line since the altitude passes through C . Next, instead of finding the midpoint of AB , find the slope of the line that is perpendicular to AB . The slope of AB is − 1/2. So, since perpendicular lines have opposite reciprocal slopes, the slope of the line perpendicular to AB is 2. Now use point-slope form to write the equation of the altitude through C .
Example 4 Using the Triangle Midsegment Theorem
A midsegment of a triangle is a segment with endpoints at the midpoints of two sides of a triangle. Every triangle has three midsegments. These three midsegments form a triangle called the midsegment triangle. By the Triangle Midsegment Theorem, if a segment is a midsegment, then the segment is parallel to a side of the triangle and its length is half the length of that side.
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