5.4 Use Medians (and Altitudes)
5.4 Use Medians and Altitudes
Goal Use medians and altitudes of triangles.
The
median of a triangle
is a segment from a vertex to the midpoint of the opposite side.
Steps
1.
Construct Δ ABC
2.
Construct midpoints of each side With all 3 sides selected from the Construct menu choose Midpoint.
3.
Construct the medians.
4.
Construct the intersection.
Select only a vertex and the midpoint across from it,
for instance select A and E, from the Construct menu
F choose Segment. Repeat to construct the other 2 medians.
A
Select 2 of the medians. Choose Construct -> Intersection.
B
D
E
C
5.
Measure segments.
Geometer’s Sketchpad Keystrokes
With the segment tool draw Δ ABC.
Select only a vertex and the intersection point. Choose
Measure -> Distance. Select the intersection point and the midpoint. Choose Measure -> Distance. (For instance measure BH then HD). Repeat to find the lengths of
F the parts of the other 2 medians. What is the relationship between each pair of measurements?
A
B
D
H
E
C
6.
Does your observed relationship persist?
Select an endpoint of your original triangle and drag it. Does your conjecture hold or should you reevaluate it?
CONCURRENCY OF MEDIANS OF A TRIANGLE
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.
The medians of Δ
ABC meet at P and AP =
3
2
_ ___, BP =
2
3
_ ___ , and OP =
2
3
_ ___.
Example
Use the centroid of a triangle
In Δ FGH , M is the centroid and GM = 6.
Find ML and GL.
The point of concurrency of the three medians of a triangle is the
CENTROID.
In the Example , suppose FM = 10.
Find MK and FK .
An
altitude of a triangle
is the perpendicular segment from a vertex to the opposite side or to the line that contains the opposite side.
For ABC
THEOREM 5.9: CONCURRENCY OF ALTITUDES OF A
TRIANGLE
Note that the altitude is also known as the “height” when finding area.
The point at which the lines containing the three altitudes of a triangle intersect is called the
ORTHOCENTER
of the triangle.
Examples: For the 3 triangles draw altitudes and the orthocenter:
Same
Triangle
Example
A
AE is an altitude.
If m
AEG = 6x - 18 and m
GEB = 3x + 18 find x.
G
If YW is an altitude of
XYZ, name a right angle.
E
B
__________________
C
Another name for an altitude is __________.
Altitudes always form __________ angles.
