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5.3 Medians and Altitudes of Triangles
The past few days we have talked about special segments of a triangle: perpendicular bisectors and angle bisectors. Today, we will study two more special segments of a triangle: medians and altitudes
A __________________________________ of a triangle is a segment who endpoints are a vertex of a triangle and the midpoint of the opposite side.
So in △ ABC shown to the right, D is the _________________________
Of side _______________ . So ________ is the median of the triangle.
The three medians of the triangle are concurrent. The point of concurrency is called the
____________________________________ of the triangle. IT IS ALWAYS INSIDE THE TRAINGLE!
The medians of a triangle have a special property. We are going to demonstrate this property in partners! Each set of partners should have a piece of paper, scissors, and a ruler.
1. Cut out a large acute, obtuse, or right triangle. Label the vertices.2. Fold the sides to locate the midpoint of each side. Label the midpoints.3. Fold to form the median from each vertex to the midpoint of the opposite side.4. Did your medians meet at about the same point? If so, label the centroid point.5. Verify the length of one median:
Length of a median(vertex to midpoint) is _____________.
Length from centroid to vertex is _____________.
5.3 Medians and Altitudes of Triangles
Example 1: Using the Centroid of a TriangleP is the centroid of △QRS shown below and PT= 5. Find RT and RP.
Solution:
OKAY! Let’s shift gears now into another segment of a triangle.
An _____________________________ of a triangle is the perpendicular segment from a vertex to the opposite side or to the line that contains the opposite side. An altitude can lay INSIDE, OUTSIDE, or ON the triangle.
Every triangle has three altitudes, and these altitudes are concurrent. They interect at a
point called the ______________________________ of the triangle.
Example 2: Drawing Altitudes and Orthocenters
Where is the orthocenter located in each type of the triangle?
a. Acute Triangle b. Right Triangle c. Obtuse Triangle
Pg 282 #3-11, 17-23