Transcript 6.3 Medians and Altitudes

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 (5th
from left, top choice).
2) Construct the altitudes from each side using the
perpendicular line tool (4th 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.