6 3 Medians and Altitudes Median of a

6. 3 Medians and Altitudes

Median of a Triangle A line segment connecting a vertex of a triangle to the midpoint of the opposite side

1) On a piece of graph paper, graph the triangle with vertices A(2, 1) B(5, 8) C(8, 3). 2) Find the midpoint of each side and label the midpoint opposite A “D”, opposite B “E”, opposite C “F”. 3) Draw the medians using a ruler. 4) Label the point of concurrency X. 5) In centimeters, find the lengths AX, BX, CX, DX, EX, FX. 6) Find a relationship between the length from the vertex to the point of concurrency and from the point of concurrency to the midpoint.

Centroid – The point of concurrency of the medians. Centroid Theorem – The centroid of a triangle is two-thirds of the distance from each vertex to the midpoint of the opposite side.

Altitude of a Triangle A perpendicular segment from a vertex to the base or to the line containing the base

Geogebra Activity 1) Construct a triangle using the polygon tool (5 th from left, top choice). 2) Construct the altitudes from each side using the perpendicular line tool (4 th from left, top choice). 3) The altitudes should be concurrent. This point of concurrency is called the orthocenter. 4) Adjust the angles of the triangle by selecting the arrow tool and changing the coordinates of the vertices of the triangle. Make a conjecture about the location of the orthocenter based of the angle classification of the triangle (obtuse vs. acute vs. right).

Orthocenter – The point of concurrency of the altitudes. If the triangle is acute, the orthocenter is inside the triangle. If the triangle is right, the orthocenter is on the triangles. If the triangle is obtuse, the orthocenter is outside the triangle.

On a piece of graph paper, graph the triangle with vertices X(-4, -1), Y(-2, 4), Z(3, -1). Find the coordinates of the orthocenter. You must use your knowledge of slopes do not just draw a line that looks perpendicular to another line.

Proof • Prove that the median from the vertex angle of an isosceles triangle to the base is also an altitude.

Copy/Complete this table in your notes Sketch Perpendicular Bisector Angle Bisector Median Altitude Point of Concurrency Property

• Answer to previous slide on p. 323 of big ideas textbook.