Medians of a Triangle
by
Nancy Perzel
LetŐs begin by defining a median.
Since there are three vertices and three sides in a triangle there are three medians.
To construct a median using GeometerŐs Sketchpad:
1) Select one side of the triangle
2) Construct a midpoint
3) Select the midpoint and opposite vertex
4) Construct a segment – this is your median
Triangle ABC has three medians.
Length of Medians
In Example 1, Triangle ABC is an isosceles triangle. AC and BC have equal lengths and ∠A ≅ ∠B.
What is the relationship between the lengths of the medians in triangle ABC?
** Opportunity for extension: Construct an equilateral triangle. What is the relationship between the lengths of the medians in an equilateral triangle? Construct a scalene triangle. What is the relationship between the lengths of the medians in a scalene triangle?
There are many explorations that can be done with the lengths of the medians. GeometerŐs Sketchpad is a great tool to use and incorporate technology.
a) In an isosceles triangle, the two medians drawn from the vertices of the equal angles are equal in length.
b) In an equilateral triangle all the medians are of the same length.
c) In a scalene triangle, all the medians are of different length.
d) The point of concurrency of the segments divides each median into two segments whose lengths are in the ratio 2:1.
** If you are interested in a proof visit: http://jwilson.coe.uga.edu/EMAT6680Fa06/Chitsonga/MEDIAN/THE%20MEDIANS%20OF%20A%20TRIANGLE.htm
Coordinate Values
When you construct all three medians of a triangle, they all intersect at the same point D. This point of concurrency is called the centroid. The centroid will always be inside the triangle.
What do we know about the coordinates of point D?
The coordinate value of point D is (4.67, 3.67)
The x-coordinate value of the centroid is equal to the mean (average) of the x-values of the three vertices of the triangle. The y-coordinate value of the centroid is equal to the mean (average) of the y-values of the three vertices of the triangle.
LetŐs see if this relationship holds true with a second example:
Before using GSP to measure the coordinates for point D, calculate the mean values for the x-coordinates and the mean values for the y-coordinates. These values should be the coordinates for the centroid, point D.
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Now, select point D and choose ŇMeasure – CoordinatesÓ to check your work! Did you get the same value?
Click here for a link to the GSP file.
On Page 10 there is an animation that can be used to vary the lengths of
segment AB for further practice.
Extension: New
Triangles Opportunity for extension -> Exploration with the
triangles formed by the mediansÉ comparing area, triangle similarity, etc. LetŐs revisit triangle XYZ from Example 2É using different
colors to highlight the interior, you can see that the three medians divide the
original triangle XYZ into six (6) smaller triangles. What relationships exist between the
new, smaller triangles? GSP helps to reveal the relationship that the three medians of a triangle divide the
triangle into six triangles that are all equal in area. If you are interested in a proof, visit: http://jwilson.coe.uga.edu/EMAT6680Fa06/Chitsonga/MEDIAN/THE%20MEDIANS%20OF%20A%20TRIANGLE.htm Are there any other similarities or relationships you can
find?
Resources: http://www.mathopenref.com/trianglecentroid.html
http://intermath.coe.uga.edu/dictnary/descript.asp?termID=214
http://intermath.coe.uga.edu/dictnary/descript.asp?termID=65 http://jwilson.coe.uga.edu/emt668/EMAT6680.Folders/McCord/Assignments/Assign4/mccord4.html
Related Investigations: http://intermath.coe.uga.edu/topics/geometry/triangle/a01.htm
http://intermath.coe.uga.edu/topics/geometry/triangle/a28.htm http://intermath.coe.uga.edu/topics/geometry/triangle/a16.htm http://intermath.coe.uga.edu/topics/geometry/triangle/a18.htm
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