Medians and Altitudes of Triangles (Special Segments)

Definitions and Theorems….. Median-- A segment whose endpoints are a vertex of the triangle and the midpoint of the opposite side. Centroid-- The point where the three medians of a triangle intersect.

Definitions and Theorems….. The centroid is located the distance from the vertex to the midpoint on the opposite side. Centroid Theorem-- Y B Centroid A D AZ = of AZ ZD = of AZ Z X C AD = of AZ

Centroid Example BG = 8 EG = 2.4 AF = 9 GD = GC = AG = BD = EC = GF = Using Triangle ABC, find the segment lengths of AG and CE. BG = 8 GD = BD = EG = 2.4 GC = EC = AF = 9 AG = GF = B E F G A D C

Construct three Medians Q (0,8) Centriod (2½, 4) R (6,4) P (2,0)

Definitions and Theorems….. Is a perpendicular segment from the vertex to the opposite side of the triangle. Altitude-- Orthocenter-- The point where the three altitudes of a triangle intersect.

Altitudes---height Altitudes of a Triangle- Orthocenter The orthocenter can be located inside, outside or on the given triangle Y B A D Z X C

Construct three altitudes Orthocenter (4.5, 3.5) Q (0,8) m = m = m = R (6,4) m = m = m = P (2,0)

Steps for Constructing Special Segments: 1. Slide 1 --- Steps for Constructing Perpendicular Bisectors 2. Slide 2 --- Steps for Constructing Angle Bisectors 3. Slide 3 --- Steps for Constructing medians and altitudes

Perpendicular Bisectors Graph the points Find the midpoint of each side Plot the midpoints Find the slope of each side Find the perpendicular slope of each side From the midpoint, count using the perpendicular slope and plot another point Draw a line segment connecting the midpoint and the point The point where all 3 perpendicular bisectors cross is called the circumcenter You will need a straight edge for this construction

Graph the points and draw a triangle Angle Bisectors Graph the points and draw a triangle Using a compass, draw an arc using one vertex as the center. This arc must pass through both sides of the angle Plot points where the arc crosses the sides of the angle From the points on the sides, create two more arcs (with the same radius) that cross. Plot a point where the two arcs cross Draw a line segment from the vertex to the point where the small arcs cross. When bisecting angles on a triangle, the point where all three angle bisectors cross is called the incenter You will need a compass and a straight edge for this construction

Graph the points for the triangle Find the midpoint of each side Medians Graph the points for the triangle Find the midpoint of each side Draw a line segment connecting the midpoint to the opposite vertex The point where the 3 medians cross is called the centroid Altitudes Graph the points for the triangle Find the slope of each side Find the perpendicular slope of each side From the opposite vertex, count using the perpendicular slope. Plot another point and draw a line segment to connect the vertex to this point The point where all 3 altitudes cross is called the orthocenter You will need a straight edge for both of these constructions