Triangle ABC has side lengths AB=8, BC=12 and CA=14. Given that D, E and F are midpoints of BC, CA and AB, respectively. Find the value of AD^2+BE^2+CF^2.
To find the value of AD^2 + BE^2 + CF^2, we can first calculate the lengths of AD, BE, and CF, and then square those lengths and sum them.
Given that D, E, and F are the midpoints of BC, CA, and AB respectively, we can use the midpoint formula to find their coordinates.
Let's assume that point A is at the origin (0, 0). Then, point B will be (8, 0) and point C will be (x, y).
Using the distance formula, we can find the coordinates of point C:
Distance BC = √((x - 8)^2 + y^2) = 12
Simplifying the equation, we get:
(x - 8)^2 + y^2 = 144 ---(1)
Since D is the midpoint of BC, its coordinates will be:
D = ((8 + x) / 2, y / 2) = (4 + x/2, y/2)
Similarly, E (the midpoint of CA) will be:
E = (x/2, y/2)
And F (the midpoint of AB) will be:
F = (8/2, 0) = (4, 0)
Now, let's find the values of x and y by substituting F coordinates (4, 0) into equation (1):
(4 - 8)^2 + (0)^2 = 144
(-4)^2 = 144
16 = 144
This is not true, and thus we can conclude that no such triangle with side lengths AB = 8, BC = 12, and CA = 14 exists.
Therefore, the value of AD^2 + BE^2 + CF^2 is not calculable.