Assignment #6 Nicole Mosteller EMAT 6680
Instructions: Construct a triangle and its medians. Construct a second triangle with the three sides having the length of the three medians from your first triangle. Find some relations between the two triangles. Prove whatever you find.
Figure 1: Triangle with constructed medians. For GSP construction script, Click Here!.
Figure 2: DEAJ is a triangle constructed from the medians of DABC.
For explanation of construction of DEAJ, Click Here!.
After investigating for relationships that exist between DEAJ and DABC, I conject that the ratio of the areas of the two triangles is 3:4.
One important fact is needed to prove this conjecture: The median of a triangle splits the triangle into two equal areas.
Lemma: The median of a triangle splits the triangle into two equal areas.
Area of DABC = Area of DABE + Area of DCAE.
Area of DABC =
.
Since AE is a median of DABC, we know that
.
Substituting this we see, Area of DABC =
.
This statement shows us that
Area of DABE = Area of DCAE.
And so the median of a triangle splits the triangle into two equal pieces.
Given: DABC with triangle constructed from its medians DEAJ. Prove: The ratio of the areas (DEAJ:DABC) = 3:4.
Because AE is a median of DABC
Area of DABC = 2(Area of DCAE).
We can see from the figure that DCAE is composed of DAEH and DCEH. Substituting this,
Area of DABC = 2(Area of DAEH + Area of DCEH).
Area of DABC = 2(Area of DAEH) + 2(Area of DCEH).
Area of DABC = Area of DEAJ + 2(Area of DCEH).
Click Here! for proof 2(Area of DAEH) = Area of DEAJ.
Area of DABC = Area of DEAJ + 2(1/2 Area of DCEF).
Click Here! for proof Area of DCEH = 1/2 Area of DCEF.
Area of DABC = Area of DEAJ + 2(1/4 Area of DCAE).
Click Here! for proof Area of DCEF= 1/2 Area of DCAE.
Area of DABC = Area of DEAJ + 2(1/8 Area of DABC).
See Lemma Above.
Area of DABC = Area of DEAJ + 1/4 Area of DABC.
Area of DABC - 1/4 Area of DABC= Area of DEAJ.
3/4 Area of DABC= Area of DEAJ.
3:4 = Area of DEAJ:Area of DABC.
Return to Nicole's Page