Problem:
Construct a triangle and its medians. Construct a second triangle with the three sides having the lengths of the three medians from your first triangle. Find some relationship between the two triangles. (E.g., are they congruent? similar? Have same area? same perimeter? ratio of areas? ratio or perimeters?) Prove whatever you find.
Investigation:
Now lets find and compare the areas of ACB triangle and DHB triangle:
Here DHB triangle congruent to YXZ triangle, which has been formed by medians of ABC triangle. In the graph as you can see DK=KH where D, K, and E are midpoints of ABC triangle.
Also we can understand from the graph KB = 3AK.
Now if we call S to the area of AKD triangle, then area of KBD triangle will be 3S.
i.e. A(AKD) = S ⇔ A(KBD) = 3S.
K is the midpoint of DH segment then A(KBD) = A(KHB) = 3S so A(DHB) = 6S.
A(AKD) = S and A(KBD) = 3S then A(ABD) = 4S. Since D is the midpoint of AC segment then A(ABD) = 4S = A(DBC), thus A(ACB) = 4S + 4S = 8S
Now the ratio of A(DHB) / A(ACB) = 6S/8S = 3/4 = 0.75. It is to check this result with GSP. Also you can see below this is the same result which we found with GSP.
Here is a GSP graph which shows relations between a triangle and its medians.
To explore this relationship you can check the GSP document :
Now lets check relation between perimeters of ACB triangle and DHB triangle:
|AF|² + |BD|² + |CE|² = |HB|² + |BD|² + |DH|², here sizes of the sides of DHB triangle are equal to sizes of the sides of ACB triangle.
= 6.27² + 11.97² + 11.53²
= 315.5347
|AB|² + |BC|² + |AC|² = 10.16² + 15.10² + 9.46²
= 420.7272
and 315.5347 ÷ 420.7272 = 0.75 = 3/4 so we can conclude
|AF|² + |BD|² + |CE|² = |HB|² + |BD|² + |DH|² = 3/4 . (|AB|² + |BC|² + |AC|²)
Also if we say k = 1/2 . (|AF| + |BD| + |CE|) = 1/2 . ( perimeter of median triangle) then
A(ACB) = 4/3. sqrt(k.(k-|AF|)(k-|BD|))(k-|CE|))
Lets prove this formula: You can see A(ACB) = 47.25 cm² from above graph.
k = 1/2 . (perimeter of median triangle) = 1/2 . (29.78) = 14.89 then
sqrt(k.(k-|AF|)(k-|BD|)(k-|CE|)) = sqrt(14.89(14.89-6.27)(14.89-11.97)(14.89-11.53))
= sqrt(1,259) = 35.4864 and 4/3. 35.4864 = 47.3 ≈ 47.25
Note:I used ten digits calculator. If you measure the size of all sides carefully then you can get the exact results.