5.4: Use Medians and Altitudes Objectives: To use and define medians and altitudes To discover and use various theorems about medians and altitudes

Special Triangle Segments Both perpendicular bisectors and angle bisectors are often associated with triangles, as shown below. Triangles have two other special segments. Perpendicular Bisector Angle Bisector

Points of Concurrency Also recall that lines are concurrent if they intersect at the same point. There are two more points of concurrency. Circumcenter: Perpendicular Bisectors Incenter: Angle Bisectors

Median

Median A median of a triangle is a segment from a vertex to the midpoint of the opposite side of the triangle.

Altitude

Altitude An altitude of a triangle is a perpendicular segment from a vertex to the opposite side or to the line that contains that side. The length of the altitude is the height of the triangle.

Example 1 Is it possible for any of the aforementioned special segments to be identical? In other words, is there a triangle for which a median, an angle bisector, and an altitude are all the same?

Intersecting Medians Activity In this activity, we are going to find the balancing point for a given triangle. Draw a triangle on a sturdy piece of paper (or wood), then cut it out.

Intersecting Medians Activity Balance the triangle on the eraser end of a pencil. Mark the balancing point on your triangle.

Intersecting Medians Activity Use a ruler to find the midpoint of each side of the triangle. Draw the three medians of the triangle.

Intersecting Medians Activity What do you notice about the point of concurrency of the three medians and the balancing point of the triangle? This point is called the centroid of the triangle.

Intersecting Medians Activity The centroid of a triangle divides each median into two parts. Use the following Geometer’s Sketchpad demonstration to see what fraction this is.

Concurrency of Medians Theorem The medians of a triangle intersect at a point that is two-thirds of the distance from each vertex to the midpoint of the opposite side.

Centroid The three medians of a triangle are concurrent. The point of concurrency is an interior point called the centroid. It is the balancing point or center of gravity of the triangle.

Example 2 In triangle RST, Q is the centroid and SQ = 8. Find QW and SW.

Another Point of Concurrency The final point of concurrency comes from the intersection of the three altitudes. Use the following Geometer’s Sketchpad activity to investigate this elusive point.

Altitudes Construct triangle ABC, then construct a line through B that is perpendicular to AC. Must an altitude always lie inside a triangle. If not, where else can it be? Construct another altitude from A to BC and F, the point of intersection of the two altitudes.

Altitudes Now construct the third altitude from C to AB. What do you notice? Re-label F as Orthocenter. Will the orthocenter always lie inside of the triangle? If not, where else can it be?

Orthocenter Concurrency of Altitudes of a Triangle Theorem The lines containing the altitudes of a triangle are concurrent. G

Orthocenter The point of concurrency of all three altitudes of a triangle is called the orthocenter of the triangle. The orthocenter, P, can be inside, on, or outside of a triangle depending on whether it is acute, right, or obtuse, respectively.

Example 3 Is it possible for any of the points of concurrency to coincide? In other words, is there a triangle for which any of the points of concurrency are the same. Use your GSP construction to determine your answer.