**Stephani Eckelkamp**

** **

**Medial Triangles**

** **

** **

**Medial triangles are formed by
connecting the midpoints of any one triangle (as seen below).**

** **

** **

**Points x, y, and z are the midpoints
of their respective sides of ****, therefore **** ****IS
THE MEDIAL TRIANGLE.**

** **

**~~~NOW THAT WE KNOW WHAT IT LOOKS LIKE LET US
LOOK AT SOME SPECIAL PROPERTIES OF MEDIAL TRIANGLES~~~**

** **

**WHAT DO WE ALREADY KNOWÉ**

** **

** **

** **

** ****WE KNOW THESE SEGMENTS ARE CONGRUENT BECAUSE THE
MIDPOINTS **

**CUT THE SEGEMTNS IN TO TWO EQUAL PARTS.
**

** **

** **

** **

**Theorem:
If a line segment joins the midpoints of two sides of a triangle, **

**Then it is parallel to the third side and equal
to one-half of it.**

** **

After applying the theorem above we get:

** **

** **

** **

**Can we demonstrate this? LetÕs look at it in the coordinate
plane.**

** **

** **

**As we can see above, the measure of
the blue triangle sides is**

**two times larger than the parallel
side formed by the medial triangle.
**

**We know that the sides formed by the
medial triangles are parallel because their slopes are equal.**

**To view this picture in GSP click here**

**Can we use the medial triangle
theorem to show that the area of the medial triangle is ¼ the area of
the original triangle?**

**Set up a two column proof to see if
you can reason this. HINT: your given will be some part of the
medial triangle theorem.**

** **

**After you have tried to work out a
proof click here for guidance
or to check your answer.**

** **

**COMPARING THE MEDIAL TRAINGLE TO THE
TRIANGLE CENTERS**

** **

**What are the triangle centers?**

** **

**O is the ****orthocenter****, which is created by the
intersection of the three altitudes of the triangle.**

** **

**I is the ****incenter****, which is the point that is equidistant from the tree sides of
the triangle.**

** **

**CC is the ****circumcenter****, which is formed from the perpendicular bisectors. The point is equidistant from the three
verticies.**

** **

**C is the ****centroid****, which is the common intersection of the three medians of the
triangle. **

** **

** **

** **

** **

** **

** **

**Triangle
centers in comparison to the medial triangle**

** **

**What are the three types of
triangles? Where do you think the
centers might be in each of these cases?**

** **

** **

__AcUTE
TRIANGLE__

** **

** **

**N****otice that most of the centers are
located inside the medial triangle.
Why isnÕt the orthocenter inside the triangle?**

** Do you think that it will ever be located inside the medial
triangle? **

** **

__RIGHT TRIANGLE__

** **

**W****hat do you notice about where the orthocenter
and the circumcenter are located?**

**The orthocenter is located on the
vertex of the right angle of the original triangle, and the circumcenter is
located on the vertex of the right angle of the medial triangle.**

** **

**Does the centroid look like it is the
center of the medial triangle?
LetÕs test this in GSP.**

** **

**We can see the centroid is the
intersection of the medians of the medial triangle and the original triangle.**

**Can you thin of possible reasons why
this is true?**

**Write down some ideas in this word document and print it out.**

**Cut and paste this picture into your
file.**

**Also create either an obtuse or
acute triangle to add your word document.
Use this link to create your
pictures.**

** **

**What do you think will happen to the
orthocenter and the circumcenter in an obtuse triangle?**

** **

__ __

__OBTUSE TRAINGLE__

__ __

** **

**Did the orthocenter and the
circumcenter move where you thought they would?**

** Explain your reasoning in your word document.**

** **

**Looking at the centers of circles is
there anything you can deduce after seeing how the centers relate **

**to one another when the triangle changes
from acute, right, and obtuse?**

** **

**For a hint click here**

** **

**Check to see if your prediction is
true in this GSP file**

** **

**Return to homepage**

** **